Construction of 2–Local Finite Groups of a Type Studied by Solomon

ISSN 1364-0380 (on line) 1465-3060 (printed) 917

Geometry & Topology Volume 6 (2002) 917–990 Published: 31 December 2002

Construction of 2–local ﬁnite groups of a type studied by Solomon and Benson Ran Levi Bob Oliver

Department of Mathematical Sciences, University of Aberdeen Meston Building 339, Aberdeen AB24 3UE, UK and LAGA – UMR 7539 of the CNRS, Institut Galil´ee Av J-B Cl´ement, 93430 Villetaneuse, France Email: [email protected] and [email protected]

Abstract

A p–local finite group is an algebraic structure with a classifying space which has many of the properties of p–completed classifying spaces of finite groups. In this paper, we construct a family of 2–local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2–subgroup of Spin7(q) (q an odd prime power) shown by Solomon not to occur as the 2–fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2–completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer–Wilkerson space BDI(4).

AMS Classiﬁcation numbers Primary: 55R35 Secondary: 55R37, 20D06, 20D20

Keywords: Classifying space, p–completion, ﬁnite groups, fusion.

Proposed:HaynesMiller Received:22October2002 Seconded: RalphCohen,BillDwyer Accepted: 31December2002

c Geometry & Topology Publications 918 Ran Levi and Bob Oliver

As one step in the classification of finite simple groups, Ron Solomon [22] considered the problem of classifying all finite simple groups whose Sylow 2–subgroups are isomorphic to those of the Conway group Co3 . The end result of his paper was that Co3 is the only such group. In the process of proving this, he needed to consider groups G in which all involutions are conjugate, and such that for any involution x ∈ G, there are subgroups K ⊳ H ⊳ CG(x) such that K and ∼ CG(x)/H have odd order and H/K = Spin7(q) for some odd prime power q. Solomon showed that such a group G does not exist. The proof of this statement was also interesting, in the sense that the 2–local structure of the group in question appeared to be internally consistent, and it was only by analyzing its interaction with the p–local structure (where p is the prime of which q is a power) that he found a contradiction. In a later paper [3], Dave Benson, inspired by Solomon’s work, constructed certain spaces which can be thought of as the 2–completed classifying spaces which the groups studied by Solomon would have if they existed. He started with the spaces BDI(4) constructed by Dwyer and Wilkerson having the property that

∗ GL4(2) H (BDI(4); F2) =∼ F2[x1,x2,x3,x4] (the rank four Dickson algebra at the prime 2). Benson then considered, for each odd prime power q, the homotopy fixed point set of the Z–action on BDI(4) generated by an “Adams operation” ψq constructed by Dwyer and Wilkerson. This homotopy fixed point set is denoted here BDI4(q). In this paper, we construct a family of 2–local finite groups, in the sense of [6], which have the 2–local structure considered by Solomon, and whose classifying spaces are homotopy equivalent to Benson’s spaces BDI4(q). The results of [6] combined with those here allow us to make much more precise the statement that these spaces have many of the properties which the 2–completed classifying spaces of the groups studied by Solomon would have had if they existed. To explain what this means, we first recall some definitions.

A fusion system over a finite p–group S is a category whose objects are the subgroups of S , and whose morphisms are monomorphisms of groups which include all those induced by conjugation by elements of S . A fusion system is saturated if it satisfies certain axioms formulated by Puig [19], and also listed in [6, Definition 1.2] as well as at the beginning of Section 1 in this paper. In particular, for any finite group G and any S ∈ Sylp(G), the category FS (G) whose objects are the subgroups of S and whose morphisms are those monomorphisms between subgroups induced by conjugation in G is a saturated fusion system over S .

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If F is a saturated fusion system over S , then a subgroup P ≤ S is called ′ ′ ′ F –centric if CS(P )= Z(P ) for all P isomorphic to P in the category F . A centric linking system associated to F consists of a category L whose objects are the F –centric subgroups of S , together with a functor L −−→F which is the inclusion on objects, is surjective on all morphism sets and which satisfies certain additional axioms (see [6, Definition 1.7]). These axioms suffice to ensure ∧ that the p–completed nerve |L|p has all of the properties needed to regard it as a “classifying space” of the fusion system F . A p–local finite group consists of a triple (S, F, L), where S is a finite p–group, F is a saturated fusion system over S , and L is a linking system associated to F . The classifying space of ∧ a p–local finite group (S, F, L) is the p–completed nerve |L|p (which is p– complete since |L| is always p–good [6, Proposition 1.12]). For example, if G is a finite group and S ∈ Sylp(G), then there is an explicitly defined centric c linking system LS(G) associated to FS(G), and the classifying space of the c c ∧ ∧ triple (S, FS (G), LS (G)) is the space |LS(G)|p ≃ BGp . Exotic examples of p–local finite groups for odd primes p — ie, examples which do not represent actual groups — have already been constructed in [6], but using ad hoc methods which seemed to work only at odd primes.

In this paper, we first construct a fusion system FSol(q) (for any odd prime power q) over a 2–Sylow subgroup S of Spin7(q), with the properties that all elements of order 2 in S are conjugate (ie, the subgroups they generated are all isomorphic in the category), and the “centralizer fusion system” (see the beginning of Section 1) of each such element is isomorphic to the fusion system of Spin7(q). We then show that FSol(q) is saturated, and has a c unique associated linking system LSol(q). We thus obtain a 2–local finite group c (S, FSol(q), LSol(q)) where by Solomon’s theorem [22] (as explained in more detail in Proposition 3.4), FSol(q) is not the fusion system of any finite group. def c ∧ c Let BSol(q) = |LSol(q)|2 denote the classifying space of (S, FSol(q), LSol(q)). ∧ Thus, BSol(q) does not have the homotopy type of BG2 for any finite group G, but does have many of the nice properties of the 2–completed classifying space of a finite group (as described in [6]).

Relating BSol(q) to BDI4(q) requires taking the “union” of the categories c n LSol(q ) for all n ≥ 1. This however is complicated by the fact that an inclusion of ﬁelds Fpm ⊆ Fpn (ie, m|n) does not induce an inclusion of cenric linking c n systems. Hence we have to replace the centric linking systems LSol(q ) by cc n the full subcategories LSol(q ) whose objects are those 2–subgroups which are c ∞ c n centric in FSol(q ) = n≥1 FSol(q ), and show that the inclusion induces a ′ n def cc n ∧ n homotopy equivalence BSSol (q ) = |LSol(q )|2 ≃ BSol(q ). Inclusions of ﬁelds

Geometry & Topology, Volume 6 (2002) 920 Ran Levi and Bob Oliver

c ∞ def do induce inclusions of these categories, so we can then deﬁne LSol(q ) = cc n n≥1 LSol(q ), and spaces

S ∞ c ∞ ∧ ′ n ∧ BSol(q )= |LSol(q )|2 ≃ BSol (q ) 2 . n[≥1 c ∞ q The category LSol(q ) has an “Adams map” ψ induced by the Frobenius au- q ∞ tomorphism x 7→ x of Fq . We then show that BSol(q ) ≃ BDI(4), the space of Dwyer and Wilkerson mentioned above; and also that BSol(q) is equivalent to the homotopy fixed point set of the Z–action on BSol(q∞) generated by q Bψ . The space BSol(q) is thus equivalent to Benson’s spaces BDI4(q) for any odd prime power q. The paper is organized as follows. Two propositions used for constructing saturated fusion systems, one very general and one more specialized, are proven in Section 1. These are then applied in Section 2 to construct the fusion systems FSol(q), and to prove that they are saturated. In Section 3 we prove the existence and uniqueness of a centric linking systems associated to FSol(q) and study their automorphisms. Also in Section 3 is the proof that FSol(q) is not the fusion system of any finite group. The connections with the space BDI(4) of Dwyer and Wilkerson is shown in Section 4. Some background material on the spinor groups Spin(V, b) over fields of characteristic 6= 2 is collected in an appendix. We would like to thank Dave Benson, Ron Solomon, and Carles Broto for their help while working on this paper.

1 Constructing saturated fusion systems

In this section, we first prove a general result which is useful for constructing saturated fusion systems. This is then followed by a more technical result, which is designed to handle the specific construction in Section 2. We first recall some definitions from [6]. A fusion system over a p–group S is a category F whose objects are the subgroups of F , such that

HomS(P,Q) ⊆ MorF (P,Q) ⊆ Inj(P,Q) for all P,Q ≤ S , and such that each morphism in F factors as the composite of an F –isomorphism followed by an inclusion. We write HomF (P,Q) = MorF (P,Q) to emphasize that the morphisms are all homomorphisms of groups.

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We say that two subgroups P,Q ≤ S are F –conjugate if they are isomorphic in F . A subgroup P ≤ S is fully centralized (fully normalized) in F if ′ ′ ′ |CS(P )| ≥ |CS(P )| (|NS(P )| ≥ |NS(P )|) for all P ≤ S which is F –conjugate to P . A saturated fusion system is a fusion system F over S which satisﬁes the following two additional conditions:

(I) For each fully normalized subgroup P ≤ S , P is fully centralized and AutS(P ) ∈ Sylp(AutF (P )).

(II) For each P ≤ S and each ϕ ∈ HomF (P,S) such that ϕ(P ) is fully centralized in F , if we set −1 Nϕ = g ∈ NS(P ) ϕcgϕ ∈ AutS(ϕ(P )) ,

then ϕ extends to a homomorphism ϕ ∈ HomF (Nϕ,S).

For example, if G is a finite group and S ∈ Sylp(G), then the category FS(G) whose objects are the subgroups of S and whose morphisms are the homomorphisms induced by conjugation in G is a saturated fusion system over S . Asub- group P ≤ S is fully centralized in FS(G) if and only if CS(P ) ∈ Sylp(CG(P )), and P is fully normalized in FS(G) if and only if NS(P ) ∈ Sylp(NG(P )). For any fusion system F over a p–group S , and any subgroup P ≤ S , the “centralizer fusion system” CF (P ) over CS(P ) is defined by setting ′ ′ ′ HomCF (P )(Q,Q )= (ϕ|Q) ϕ ∈ HomF (PQ,PQ ), ϕ(Q) ≤ Q , ϕ|P = IdP ′ for all Q,Q ≤ CS(P ) (see [6, Definition A.3] or [19] for more detail). We also write CF (g) = CF (hgi) for g ∈ S . If F is a saturated fusion system and P is fully centralized in F , then CF (P ) is saturated by [6, Proposition A.6] (or [19]).

Proposition 1.1 Let F be any fusion system over a p–group S . Then F is saturated if and only if there is a set X of elements of order p in S such that the following conditions hold:

(a) Each x ∈ S of order p is F –conjugate to some element of X. (b) If x and y are F –conjugate and y ∈ X, then there is some morphism ψ ∈ HomF (CS(x),CS (y)) such that ψ(x)= y.

Proof Throughout the proof, conditions (I) and (II) always refer to the conditions in the deﬁnition of a saturated fusion system, as stated above or in [6, Deﬁnition 1.2].

Geometry & Topology, Volume 6 (2002) 922 Ran Levi and Bob Oliver

Assume first that F is saturated, and let X be the set of all x ∈ S of order p such that hxi is fully centralized. Then condition (a) holds by definition, (b) follows from condition (II), and (c) holds by [6, Proposition A.6] or [19]. Assume conversely that X is chosen such that conditions (a–c) hold for F . Define

T U = (P, x) P ≤ S, |x| = p, x ∈ Z(P ) , some T ∈ Sylp(AutF (P )), T ≥ AutS(P ) ,

T where Z(P ) is the subgroup of elements of Z(P ) fixed by the action of T . Let U0 ⊆ U be the set of pairs (P,x) such that x ∈ X. For each 1 6= P ≤ S , there is some x such that (P,x) ∈U (since every action of a p–group on Z(P ) has nontrivial fixed set); but x need not be unique. We first check that

(P,x) ∈U0, P fully centralized in CF (x) =⇒ P fully centralized in F . (1)

Assume otherwise: that (P,x) ∈ U0 and P is fully centralized in CF (x), but ′ ′ P is not fully centralized in F . Let P ≤ S and ϕ ∈ IsoF (P, P ) be such ′ ′ ′ that |CS(P )| < |CS(P )|. Set x = ϕ(x) ≤ Z(P ). By (b), there exists ψ ∈ ′ ′ ′′ ′ HomF (CS (x ),CS (x)) such that ψ(x ) = x. Set P = ψ(P ). Then ψ ◦ ϕ ∈ ′′ ′′ IsoCF (x)(P, P ), and in particular P is CF (x)–conjugate to P . Also, since ′ ′ ′ ′′ CS(P ) ≤ CS(x ), ψ sends CS(P ) injectively into CS(P ), and |CS(P )| < ′ ′′ ′′ ′′ |CS(P )| ≤ |CS(P )|. Since CS(P ) = CCS (x)(P ) and CS(P ) = CCS (x)(P ), this contradicts the original assumption that P is fully centralized in CF (x).

By deﬁnition, for each (P,x) ∈ U , NS(P ) ≤ CS(x) and hence AutCS (x)(P ) = AutS(P ). By assumption, there is T ∈ Sylp(AutF (P )) such that τ(x)= x for all τ ∈ T ; ie, such that T ≤ AutCF (x)(P ). In particular, it follows that

∀(P, x) ∈U : AutS(P ) ∈ Sylp(AutF (P )) ⇐⇒ AutCS (x)(P ) ∈ Sylp(AutCF (x)(P )). (2) We are now ready to prove condition (I) for F ; namely, to show for each P ≤ S fully normalized in F that P is fully centralized and AutS(P ) ∈ ′ ′ Sylp(AutF (P )). By deﬁnition, |NS(P )| ≥ |NS(P )| for all P F –conjugate to P . Choose x ∈ Z(P ) such that (P,x) ∈U ; and let T ∈ Sylp(AutF (P )) be T such that T ≥ AutS(P ) and x ∈ Z(P ) . By (a) and (b), there is an element y ∈ X and a homomorphism ψ ∈ HomF (CS (x),CS(y)) such that ψ(x) = y. ′ ′ −1 ′ Set P = ψP , and set T = ψT ψ ∈ Sylp(AutF (T )). Since T ≥ AutS(P ) by ′ deﬁnition of U , and ψ(NS(P )) = NS(P ) by the maximality assumption, we ′ ′ ′ T ′ ′ see that T ≥ AutS(P ). Also, y ∈ Z(P ) (T y = y since Tx = x), and this ′ ′ ′ shows that (P ,y) ∈ U0 . The maximality of |NS(P )| = |NCS (y)(P )| implies ′ that P is fully normalized in CF (y). Hence by condition (I) for the saturated

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fusion system CF (y), together with (1) and (2), P fully centralized in F and AutS(P ) ∈ Sylp(AutF (P )).

It remains to prove condition (II) for F . Fix 1 6= P ≤ S and ϕ ∈ HomF (P,S) def such that P ′ = ϕP is fully centralized in F , and set

−1 ′ Nϕ = g ∈ NS(P ) ϕcgϕ ∈ AutS(P ) . ′ We must show that ϕ extends to some ϕ ∈ HomF (Nϕ,S). Choose some x ∈ ′ ′ Z(P ) of order p which is ﬁxed under the action of AutS(P ), and set x = −1 ′ −1 ′ ′ ϕ (x ) ∈ Z(P ). For all g ∈ Nϕ , ϕcgϕ ∈ AutS(P ) ﬁxes x , and hence cg(x)= x. Thus ′ ′ x ∈ Z(Nϕ) and hence Nϕ ≤ CS(x); and NS(P ) ≤ CS(x ). (3) Fix y ∈ X which is F –conjugate to x and x′ , and choose

′ ′ ψ ∈ HomF (CS(x),CS (y)) and ψ ∈ HomF (CS(x ),CS(y)) such that ψ(x)= ψ′(x′)= y. Set Q = ψ(P ) and Q′ = ψ′(P ′). Since P ′ is fully ′ ′ ′ ′ ′ centralized in F , ψ (P )= Q , and CS(P ) ≤ CS(x ), we have

′ ′ ′ ′ ′ ′ ′ ψ (CCS (x )(P )) = ψ (CS(P )) = CS(Q )= CCS(y)(Q ). (4) ′ −1 ′ Set τ = ψ ϕψ ∈ IsoF (Q,Q ). By construction, τ(y) = y, and thus τ ∈ ′ ′ ′ IsoCF (y)(Q,Q ). Since P is fully centralized in F , (4) implies that Q is fully centralized in CF (y). Hence condition (II), when applied to the saturated fusion system CF (y), shows that τ extends to a homomorphism τ ∈

HomCF (y)(Nτ ,CS (y)), where

−1 ′ Nτ = g ∈ NCS (y)(Q) τcgτ ∈ AutCS (y)(Q ) . Also, for all g ∈ Nϕ ≤ CS(x) (see (3)),

−1 −1 ′ ′ −1 ′ ′ cτ (ψ(g)) = τcψ(g)τ = (τψ)cg(τψ) = (ψ ϕ)cg(ψ ϕ) = cψ (h) ∈ AutCS (y)(Q )

′ −1 for some h ∈ NS(P ) such that ϕcgϕ = ch . This shows that ψ(Nϕ) ≤ Nτ ; ′ ′ ′ and also (since CS(Q )= ψ (CS(P )) by (4)) that

′ ′ ′ τ(ψ(Nϕ)) ≤ ψ (NCS (x )(P )). We can now deﬁne

def ′ −1 ϕ = (ψ ) ◦ (τ ◦ ψ)|Nϕ ∈ HomF (Nϕ,S), and ϕ|P = ϕ.

Geometry & Topology, Volume 6 (2002) 924 Ran Levi and Bob Oliver

Proposition 1.1 will also be applied in a separate paper of Carles Broto and Jesper Møller [7] to give a construction of some “exotic” p–local finite groups at certain odd primes. Our goal now is to construct certain saturated fusion systems, by starting with the fusion system of Spin7(q) for some odd prime power q, and then adding to that the automorphisms of some subgroup of Spin7(q). This is a special case of the general problem of studying fusion systems generated by fusion subsystems, and then showing that they are saturated. We first fix some notation. If F1 and F2 are two fusion systems over the same p–group S , then hF1, F2i denotes the fusion system over S generated by F1 and F2 : the smallest fusion system over S which contains both F1 and F2 . More generally, if F is a fusion system over S , and F0 is a fusion system over a subgroup S0 ≤ S , then hF; F0i denotes the fusion system over S generated by the morphisms in F between subgroups of S , together with morphisms in F0 between subgroups of S0 only. In other words, a morphism in hF; F0i is a composite

ϕ1 ϕ2 ϕk P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−→ Pk, where for each i, either ϕi ∈ HomF (Pi−1, Pi), or ϕi ∈ HomF0 (Pi−1, Pi) (and Pi−1, Pi ≤ S0 ).

As usual, when G is a ﬁnite group and S ∈ Sylp(G), then FS(G) denotes the fusion system of G over S . If Γ ≤ Aut(G) is a group of automorphisms which contains Inn(G), then FS(Γ) will denote the fusion system over S whose morphisms consist of all restrictions of automorphisms in Γ to monomorphisms between subgroups of S . The next proposition provides some fairly specialized conditions which imply that the fusion system generated by the fusion system of a group G together with certain automorphisms of a subgroup of G is saturated.

Proposition 1.2 Fix a ﬁnite group G, a prime p dividing |G|, and a Sylow ⊳ p–subgroup S ∈ Sylp(G). Fix a normal subgroup Z G of order p, an elementary abelian subgroup U ⊳ S of rank two containing Z such that CS(U) ∈ Sylp(CG(U)), and a subgroup Γ ≤ Aut(CG(U)) containing Inn(CG(U)) such that γ(U)= U for all γ ∈ Γ. Set

def S0 = CS(U) and F = hFS(G); FS0 (Γ)i, and assume the following hold.

(a) All subgroups of order p in S diﬀerent from Z are G–conjugate.

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(b) Γ permutes transitively the subgroups of order p in U .

(c) {ϕ ∈ Γ | ϕ(Z)= Z} = AutNG(U)(CG(U)). (d) For each E ≤ S which is elementary abelian of rank three, contains U , and is fully centralized in FS(G),

{α ∈ AutF (CS(E)) | α(Z)= Z} = AutG(CS(E)). (e) For all E, E′ ≤ S which are elementary abelian of rank three and contain U , if E and E′ are Γ–conjugate, then they are G–conjugate.

Then F is a saturated fusion system over S . Also, for any P ≤ S such that Z ≤ P , {ϕ ∈ HomF (P,S) | ϕ(Z)= Z} = HomG(P,S). (1)

Proposition 1.2 follows from the following three lemmas. Throughout the proofs of these lemmas, references to points (a–e) mean to those points in the hypotheses of the proposition, unless otherwise stated.

Lemma 1.3 Under the hypotheses of Proposition 1.2, for any P ≤ S and any central subgroup Z′ ≤ Z(P ) of order p, ′ ′ Z 6= Z ≤ U =⇒ ∃ ϕ ∈ HomΓ(P,S0) such that ϕ(Z )= Z (1) and ′ ′ Z U =⇒ ∃ ψ ∈ HomG(P,S0) such that ψ(Z ) ≤ U . (2)

Proof Note ﬁrst that Z ≤ Z(S), since it is a normal subgroup of order p in a p–group. Assume Z 6= Z′ ≤ U . Then U = ZZ′ , and ′ ′ P ≤ CS(Z )= CS(ZZ )= CS(U)= S0 since Z′ ≤ Z(P ) by assumption. By (b), there is α ∈ Γ such that α(Z′)= Z . −1 Since S0 ∈ Sylp(CG(U)), there is h ∈ CG(U) such that h·α(P )·h ≤ S0 ; and since

ch ∈ AutNG(U)(CG(U)) ≤ Γ

def ′ by (c), ϕ = ch ◦ α ∈ HomΓ(P,S0) and sends Z to Z . If Z′ U , then by (a), there is g ∈ G such that gZ′g−1 ≤ UrZ . Since Z is central in S , gZ′g−1 is central in gP g−1 , and U is generated by Z and gZ′g−1 , −1 it follows that gP g ≤ CG(U). Since S0 ∈ Sylp(CG(U)), there is h ∈ CG(U) −1 −1 such that h(gP g )h ≤ S0 ; and we can take ψ = chg ∈ HomG(P,S0).

Geometry & Topology, Volume 6 (2002) 926 Ran Levi and Bob Oliver

We are now ready to prove point (1) in Proposition 1.2.

Lemma 1.4 Assume the hypotheses of Proposition 1.2, and let

F = hFS(G); FS0 (Γ)i be the fusion system generated by G and Γ. Then for all P, P ′ ≤ S which contain Z , ′ ′ {ϕ ∈ HomF (P, P ) | ϕ(Z)= Z} = HomG(P, P ).

Proof Upon replacing P ′ by ϕ(P ) ≤ P ′ , we can assume that ϕ is an isomorphism, and thus that it factors as a composite of isomorphisms ϕ1 ϕ2 ϕ3 ϕk−1 ϕk ′ P = P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−→ Pk = P , =∼ =∼ =∼ =∼ =∼ where for each i, ϕi ∈ HomG(Pi−1, Pi) or ϕi ∈ HomΓ(Pi−1, Pi). Let Zi ≤ Z(Pi) be the subgroups of order p such that Z0 = Zk = Z and Zi = ϕi(Zi−1). To simplify the discussion, we say that a morphism in F is of type (G) if it is given by conjugation by an element of G, and of type (Γ) if it is the restriction of an automorphism in Γ. More generally, we say that a morphism is of type (G, Γ) if it is the composite of a morphism of type (G) followed by one of type (Γ), etc. We regard IdP , for all P ≤ S , to be of both types, even if P S0 . By deﬁnition, if any nonidentity isomorphism is of type (Γ), then its source and image are both contained in S0 = CS(U).

For each i, using Lemma 1.3, choose some ψi ∈ HomF (PiU, S) such that ψi(Zi) = Z . More precisely, using points (1) and (2) in Lemma 1.3, we can choose ψi to be of type (Γ) if Zi ≤ U (the inclusion if Zi = Z ), and to be ′ of type (G, Γ) if Z U . Set Pi = ψi(Pi). To keep track of the eﬀect of morphisms on the subgroups Zi , we write them as morphisms between pairs, as shown below. Thus, ϕ factors as a composite of isomorphisms − ψ 1 ′ i−1 ϕi ψi ′ (Pi−1,Z) −−−−−→ (Pi−1,Zi−1) −−−−−→ (Pi,Zi) −−−−−→ (Pi ,Z).

If ϕi is of type (G), then this composite (after replacing adjacent morphisms of the same type by their composite) is of type (Γ, G, Γ). If ϕi is of type (Γ), then the composite is again of type (Γ, G, Γ) if either Zi−1 ≤ U or Zi ≤ U , and is of type (Γ, G, Γ, G, Γ) if neither Zi−1 nor Zi is contained in U . So we are reduced to assuming that ϕ is of one of these two forms. Case 1 Assume ﬁrst that ϕ is of type (Γ, G, Γ); ie, a composite of isomorphisms of the form ϕ1 ϕ2 ϕ3 (P0,Z) −−−−→ (P1,Z1) −−−−→ (P2,Z2) −−−−→ (P3,Z). (Γ) (G) (Γ)

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Then Z1 = Z if and only if Z2 = Z because ϕ2 is of type (G). If Z1 = Z2 = Z , then ϕ1 and ϕ3 are of type (G) by (c), and the result follows.

If Z1 6= Z 6= Z2 , then U = ZZ1 = ZZ2 , and thus ϕ2(U) = U . Neither ϕ1 nor ϕ3 can be the identity, so Pi ≤ S0 = CS(U) for all i by deﬁnition of HomΓ(−, −), and hence ϕ2 is of type (Γ) by (c). It follows that ϕ ∈ IsoΓ(P0, P3) sends Z to itself, and is of type (G) by (c) again. Case 2 Assume now that ϕ is of type (Γ, G, Γ, G, Γ); more precisely, that it is a composite of the form

ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 (P0,Z) −−−→ (P1,Z1) −−−→ (P2,Z2) −−−→ (P3,Z3) −−−→ (P4,Z4) −−−→ (P5,Z), (Γ) (G) (Γ) (G) (Γ) where Z2,Z3 U . Then Z1,Z4 ≤ U and are distinct from Z , and the groups P0, P1, P4, P5 all contain U since ϕ1 and ϕ5 (being of type (Γ)) leave U invariant. In particular, P2 and P3 contain Z , since P1 and P4 do and ϕ2, ϕ4 are of type (G). We can also assume that U ≤ P2, P3 , since otherwise P2 ∩ U = Z or P3 ∩ U = Z , ϕ3(Z)= Z , and hence ϕ3 is of type (G) by (c) again. Finally, we assume that P2, P3 ≤ S0 = CS(U), since otherwise ϕ3 = Id.

Let Ei ≤ Pi be the rank three elementary abelian subgroups deﬁned by the requirements that E2 = UZ2 , E3 = UZ3 , and ϕi(Ei−1) = Ei . In particular, Ei ≤ Z(Pi) for i = 2, 3 (since Zi ≤ Z(Pi), and U ≤ Z(Pi) by the above remarks); and hence Ei ≤ Z(Pi) for all i. Also, U = ZZ4 ≤ ϕ4(E3)= E4 since ϕ4(Z)= Z , and thus U = ϕ5(U) ≤ E5 . Via similar considerations for E0 and E1 , we see that U ≤ Ei for all i.

Set H = CG(U) for short. Let E3 be the set of all elementary abelian subgroups E ≤ S of rank three which contain U , and with the property that CS(E) ∈ Sylp(CH (E)). Since CS(E) ≤ CS(U) = S0 ∈ Sylp(H), the last condition implies that E is fully centralized in the fusion system FS0 (H). If E ≤ S is any rank three elementary abelian subgroup which contains U , then there ′ −1 is some a ∈ H such that E = aEa ∈ E3 , since FS0 (H) is saturated and ′ ′ U ⊳ H . Then ca ∈ IsoG(E, E ) ∩ IsoΓ(E, E ) by (c). So upon composing with such isomorphisms, we can assume that Ei ∈ E3 for all i, and also that ϕi(CS (Ei−1)) = CS(Ei) for each i.

In this way, ϕ can be assumed to extend to an F –isomorphism ϕ from CS(E0) to CS(E5) which sends Z to itself. By (e), the rank three subgroups Ei are −1 all G–conjugate to each other. Choose g ∈ G such that gE5g = E0 . Then −1 g·CS(E5)·g and CS(E0) are both Sylow p–subgroups of CG(E0), so there −1 is h ∈ CG(E0) such that (hg)CS (E5)(hg) = CS(E0). By (d), chg ◦ ϕ ∈ AutF (CS(E0)) is of type (G); and thus ϕ ∈ IsoG(P0, P5).

Geometry & Topology, Volume 6 (2002) 928 Ran Levi and Bob Oliver

To ﬁnish the proof of Proposition 1.2, it remains only to show:

Lemma 1.5 Under the hypotheses of Proposition 1.2, the fusion system F generated by FS(G) and FS0 (Γ) is saturated.

Proof We apply Proposition 1.1, by letting X be the set of generators of Z . Condition (a) of the proposition (every x ∈ S of order p is F –conjugate to an element of X) holds by Lemma 1.3. Condition (c) holds since CF (Z) is the fusion system of the group CG(Z) by Lemma 1.4, and hence is saturated by [6, Proposition 1.3]. It remains to prove condition (b) of Proposition 1.1. We must show that if y, z ∈ S are F –conjugate and hzi = Z , then there is ψ ∈ HomF (CS(y),CS(z)) such that ψ(y)= z. If y∈ / U , then by Lemma 1.3(2), there is ϕ ∈ HomF (CS(y),S0) such that ϕ(y) ∈ U . If y ∈ UrZ , then by Lemma 1.3(1), there is ϕ ∈ HomF (CS (y),S0) such that ϕ(y) ∈ Z . We are thus reduced to the case where y, z ∈ Z (and are F –conjugate). In this case, then by Lemma 1.4, there is g ∈ G such that z = gyg−1 . Since −1 Z ⊳ G, [G:CG(Z)] is prime to p, so S and gSg are both Sylow p–subgroups of CG(Z), and hence are CG(Z)–conjugate. We can thus choose g such that −1 −1 z = gyg and gSg = S . Since CS(y) = CS(z) = S (Z ≤ Z(S) since it is a normal subgroup of order p), this shows that cg ∈ IsoG(CS(y),CS (z)), and ﬁnishes the proof of (b) in Proposition 1.1.

2 A fusion system of a type considered by Solomon

The main result of this section and the next is the following theorem:

Theorem 2.1 Let q be an odd prime power, and ﬁx S ∈ Syl2(Spin7(q)). Let z ∈ Z(Spin7(q)) be the central element of order 2. Then there is a saturated fusion system F = FSol(q) which satisﬁes the following conditions:

(a) CF (z)= FS(Spin7(q)) as fusion systems over S . (b) All involutions of S are F –conjugate.

c Furthermore, there is a unique centric linking system L = LSol(q) associated to F .

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Theorem 2.1 will be proven in Propositions 2.11 and 3.3. Later, at the end of Section 3, we explain why Solomon’s theorem [22] implies that these fusion systems are not the fusion systems of any finite groups, and hence that the spaces BSol(q) are not homotopy equivalent to the 2–completed classifying spaces of any finite groups. Background results needed for computations in Spin(V, b) have been collected in Appendix A. We focus attention here on SO7(q) and Spin7(q). In fact, since we want to compare the constructions over Fq with those over its field extensions, most of the constructions will first be made in the groups SO7(Fq) and Spin7(Fq). We now fix, for the rest of the section, an odd prime power q. It will be ∞ def convenient to write Spin7(q ) = Spin7(Fq), etc. In order to make certain computations more explicit, we set 0 ∼ 7 b V∞ = M2(Fq) ⊕ M2 (Fq) = (Fq) and (A, B) = det(A) + det(B) 0 (where M2 (−) is the group of (2×2) matrices of trace zero), and for each n ≥ 1 0 b set Vn = M2(Fqn ) ⊕ M2 (Fqn ) ⊆ V∞ . Then is a nonsingular quadratic form ∞ n on V∞ and on Vn . Identify SO7(q )= SO(V∞, b) and SO7(q )= SO(Vn, b), n ∞ and similarly for Spin7(q ) ≤ Spin7(q ). For all α ∈ Spin(M2(Fq), det) and 0 ∞ β ∈ Spin(M2 (Fq), det), we write α ⊕ β for their image in Spin7(q ) under the natural homomorphism

∞ ∞ ∞ ι4,3 : Spin4(q ) × Spin3(q ) −−−−−→ Spin7(q ). There are isomorphisms

∞ ∞ =∼ ∞ ∞ =∼ ∞ ρ4 : SL2(q ) × SL2(q ) −−→ Spin4(q ) and ρ3 : SL2(q ) −−→ Spin3(q ) which are deﬁned explicitly in Proposition A.5, and which restrict to isomor- e e phisms n n ∼ n n ∼ n SL2(q ) × SL2(q ) = Spin4(q ) and SL2(q ) = Spin3(q ) for each n. Let

z = ρ4(−I, −I) ⊕ 1 = 1 ⊕ ρ3(−I) ∈ Z(Spin7(q)) denote the central element of order two, and set e e z1 = ρ4(−I,I) ⊕ 1 ∈ Spin7(q).

Here, 1 ∈ Spin (q) (k = 3, 4) denotes the identity element. Deﬁne U = hz, z1i. k e

Geometry & Topology, Volume 6 (2002) 930 Ran Levi and Bob Oliver

Deﬁnition 2.2 Deﬁne ∞ 3 ∞ ω : SL2(q ) −−−−−→ Spin7(q ) by setting ω(A1, A2, A3)= ρ4(A1, A2) ⊕ ρ3(A3) ∞ for A1, A2, A3 ∈ SL2(q ). Set ∞ ∞ 3 e e H(q )= ω(SL2(q ) ) and [[A1, A2, A3]] = ω(A1, A2, A3) .

Since ρ3 and ρ4 are isomorphisms, Ker(ω) = Ker(ι4,3), and thus Ker(ω)= h(−I, −I, −I)i. e e ∞ ∞ 3 In particular, H(q ) =∼ (SL2(q ) )/{±(I,I,I)}. Also,

z = [[I,I, −I]] and z1 = [[−I,I,I]], and thus U = [[±I, ±I, ±I]] (with all combinations of signs). For each 1 ≤ n< ∞, the natural homomorphism n n Spin7(q ) −−−−−−→ SO7(q ) has kernel and cokernel both of order 2. The image of this homomorphism n n is the commutator subgroup Ω7(q ) ⊳ SO7(q ), which is partly described by Lemma A.4(a). In contrast, since all elements of Fq are squares, the natural ∞ ∞ homomorphism from Spin7(q ) to SO7(q ) is surjective.

Lemma 2.3 There is an element τ ∈ N (U) of order 2 such that Spin7(q) −1 τ·[[A1, A2, A3]]·τ = [[A2, A1, A3]] (1) ∞ for all A1, A2, A3 ∈ SL2(q ).

Proof Let τ ∈ SO7(q) be the involution defined by setting τ(X,Y ) = (−θ(X), −Y ) 0 for (X,Y ) ∈ V∞ = M2(Fq) ⊕ M2 (Fq), where a b d −b θ c d = −c a . ∞ Let τ ∈ Spin7(q ) be a lifting of τ . The (−1)–eigenspace of τ on V∞ has orthogonal basis 1 0 0 1 0 1 (I, 0) , 0, 0 −1 , 0, 1 0 , 0, −1 0 , Geometry & Topology, Volume 6 (2002) Construction of 2–local finite groups 931 and in particular has discriminant 1 with respect to this basis. Hence by Lemma A.4(a), τ ∈ Ω7(q), and so τ ∈ Spin7(q). Since in addition, the (−1)–eigenspace of τ is 4–dimensional, Lemma A.4(b) applies to show that τ 2 = 1. ∞ By definition of the isomorphisms ρ3 and ρ4 , for all Ai ∈ SL2(q ) (i = 1, 2, 3) and all (X,Y ) ∈ V∞ , e e −1 −1 [[A1, A2, A3]](X,Y ) = (A1XA2 , A3Y A3 ). ∞ ∞ Here, Spin7(q ) acts on V∞ via its projection to SO7(q ). Also, for all X,Y ∈ M2(Fq), 0 1 t 0 1 −1 θ(X)= −1 0 ·X · −1 0 and in particular θ(XY )= θ(Y )·θ(X); −1 ∞ and θ(X) = X if det(X) = 1. Hence for all A1, A2, A3 ∈ SL2(q ) and all (X,Y ) ∈ V∞ , −1 −1 −1 τ·[[A1, A2, A3]]·τ (X,Y )= τ(−A1·θ(X)·A2 , −A3Y A3 ) −1 −1 = (A2XA1 , A3Y A3 ) = [[A2, A1, A3]](X,Y ). ∞ This shows that (1) holds modulo hzi = Z(Spin7(q )). We thus have two ∞ ∞ 3 automorphisms of H(q ) =∼ (SL2(q ) )/{±(I,I,I)} — conjugation by τ and the permutation automorphism — which are liftings of the same automorphism of H(q∞)/hzi. Since H(q∞) is perfect, each automorphism of H(q∞)/hzi has at most one lifting to an automorphism of H(q∞), and thus (1) holds. Also, since U is the subgroup of all elements [[±I, ±I, ±I]] with all combinations of signs, formula (1) shows that τ ∈ N (U). Spin7(q) Definition 2.4 For each n ≥ 1, set n ∞ n n n 3 n H(q )= H(q ) ∩ Spin7(q ) and H0(q )= ω(SL2(q ) ) ≤ H(q ). Define n n Γn = Inn(H(q )) ⋊ Σ3 ≤ Aut(H(q )), where Σ3 denotes the group of permutation automorphisms b n Σ3 = [[A1, A2, A3]] 7→ [[Aσ1, Aσ2, Aσ3]] σ ∈ Σ3 ≤ Aut(H(q )) . b qn ∞ For eachb n, let ψ be the automorphism of Spin7(q ) induced by the field pn n isomorphism (q 7→ q ). By Lemma A.3, Spin7(q ) is the fixed subgroup of qn n ψ . Hence each element of H(q ) is of the form [[A1, A2, A3]], where either n n qn Ai ∈ SL2(q ) for each i (and the element lies in H0(q )), or ψ (Ai) = −Ai n n for each i. This shows that H0(q ) has index 2 in H(q ). n n The goal is now to choose compatible Sylow subgroups S(q ) ∈ Syl2(Spin7(q )) n n (all n ≥ 1) contained in N(H(q )), and let FSol(q ) be the fusion system over n n S(q ) generated by conjugation in Spin7(q ) and by restrictions of Γn .

Geometry & Topology, Volume 6 (2002) 932 Ran Levi and Bob Oliver

Proposition 2.5 The following hold for each n ≥ 1.

n n (a) H(q )= CSpin7(q )(U). n n (b) N n (U) = N n (H(q )) = H(q )·hτi, and contains a Sylow Spin7(q ) Spin7(q ) n 2–subgroup of Spin7(q ).

Proof Let z1 ∈ SO7(q) be the image of z1 ∈ Spin7(q). Set V− = M2(Fq) and 0 V+ = M2 (Fq): the eigenspaces of z1 acting on V . By Lemma A.4(c),

C ∞ (U)= C ∞ (z ) Spin7(q ) Spin7(q ) 1 ∞ ∞ is the group of all elements α ∈ Spin7(q ) whose image α ∈ SO7(q ) has the form α = α− ⊕ α+ where α± ∈ SO(V±). In other words, ∞ ∞ ∞ 3 ∞ C ∞ (U)= ι Spin (q ) × Spin (q ) = ω(SL (q ) )= H(q ). Spin7(q ) 4,3 4 3 2 Furthermore, since −1 −1 τz1τ = τ[[−I,I,I]]τ = [[I, −I,I]] = zz1

∞ by Lemma 2.3, and since any element of NSpin7(q )(U) centralizes z, conjuga- ∞ tion by τ generates OutSpin7(q )(U). Hence ∞ N ∞ (U)= H(q )·hτi. Spin7(q ) Point (a), and the ﬁrst part of point (b), now follow upon taking intersections n with Spin7(q ). n If N n (U) did not contain a Sylow 2–subgroup of Spin (q ), then since Spin7(q ) 7 n every noncentral involution of Spin7(q ) is conjugate to z1 (Proposition A.8), the Sylow 2–subgroups of Spin7(q) would have no normal subgroup isomorphic 2 to C2 . By a theorem of Hall (cf [15, Theorem 5.4.10]), this would imply that they are cyclic, dihedral, quaternion, or semidihedral. This is clearly not the

n case, so NSpin7(q )(U) must contain a Sylow 2–subgroup of Spin7(q), and this ﬁnishes the proof of point (b). Alternatively, point (b) follows from the standard formulas for the orders of these groups (cf [24, pages 19,140]), which show that |Spin (qn)| q9n(q6n − 1)(q4n − 1)(q2n − 1) q2n + 1 7 = = q6n(q4n + q2n + 1) |H(qn)·hτi| 2·[qn(q2n − 1)]3 2 is odd.

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 933

n We next ﬁx, for each n, a Sylow 2–subgroup of Spin7(q ) which is contained n in H(q )·hτi = N n (U). Spin7(q )

Deﬁnition 2.6 Fix elements A, B ∈ SL2(q) such that hA, Bi =∼ Q8 (a quaternion group of order 8), and set A = [[A, A, A]] and B = [[B,B,B]]. Let ∞ ∞ C(q ) ≤ CSL2(q )(A) be the subgroup of elements of 2–power order in the centralizer (which is abelian), and setb Q(q∞)= hC(q∞),Bb i. Deﬁne ∞ ∞ 3 ∞ S0(q )= ω(Q(q ) ) ≤ H0(q ) and ∞ ∞ ∞ ∞ S(q )= S0(q )·hτi≤ H(q ) ≤ Spin7(q ).

Here, τ ∈ Spin7(q) is the element of Lemma 2.3. Finally, for each n ≥ 1, deﬁne

n ∞ n n ∞ n C(q )= C(q ) ∩ SL2(q ), Q(q )= Q(q ) ∩ SL2(q ), n ∞ n n ∞ n S0(q )= S0(q ) ∩ Spin7(q ), and S(q )= S(q ) ∩ Spin7(q ).

∞ Since the two eigenvalues of A are distinct, its centralizer in SL2(q ) is conjugate to the subgroup of diagonal matrices, which is abelian. Thus C(q∞) is conjugate to the subgroup of diagonal matrices of 2–power order. This shows that each ﬁnite subgroup of C(q∞) is cyclic, and that each ﬁnite subgroup of Q(q∞) is cyclic or quaternion.

n n Lemma 2.7 For all n, S(q ) ∈ Syl2(Spin7(q )).

Proof By [23, 6.23], A is contained in a cyclic subgroup of order qn − 1 or qn + 1 (depending on which of them is divisible by 4). Also, the normalizer of this cyclic subgroup is a quaternion group of order 2(qn ± 1), and the formula n n 2n |SL2(q )| = q (q − 1) shows that this quaternion group has odd index. Thus n n n 3 by construction, Q(q ) is a Sylow 2–subgroup of SL2(q ). Hence ω(Q(q ) ) is n ∞ 3 n a Sylow 2–subgroup of H0(q ), so ω(Q(q ) )∩Spin7(q ) is a Sylow 2–subgroup of H(qn). It follows that S(qn) is a Sylow 2–subgroup of H(qn)·hτi, and hence n also of Spin7(q ) by Proposition 2.5(b).

Following the notation of Deﬁnition A.7, we say that an elementary abelian n 2–subgroup E ≤ Spin7(q ) has type I if its eigenspaces all have square discriminant, and has type II otherwise. Let Er be the set of elementary abelian n I II subgroups of rank r in Spin7(q ) which contain z, and let Er and Er be the sets of those of type I or II, respectively. In Proposition A.8, we show that I there are two conjugacy classes of subgroups in E4 and one conjugacy class of

Geometry & Topology, Volume 6 (2002) 934 Ran Levi and Bob Oliver

II subgroups in E4 . In Proposition A.9, an invariant xC(E) ∈ E is deﬁned, for I all E ∈ E4 (and where C is one of the conjugacy classes in E4 ) as a tool for determining the conjugacy class of a subgroup. More precisely, E has type I if and only if xC(E) ∈ hzi, and E ∈C if and only if xC(E) = 1. The next lemma provides some more detailed information about the rank four subgroups and these invariants. Recall that we deﬁne A = [[A, A, A]] and B = [[B,B,B]].

n Lemma 2.8 Fix n ≥b 1, set E∗ = hz, zb1, A, Bi ≤ S(q ), and let C be the n U Spin7(q )–conjugacy class of E∗ . Let E4 be the set of all elementary abelian n subgroups E ≤ S(q ) of rank 4 which containb b U = hz, z1i. Fix a generator n 2n n X ∈ C(q ) (the 2–power torsion in CSL2(q )(A)), and choose Y ∈ C(q ) such that Y 2 = X . Then the following hold.

(a) E∗ has type I. U ′ (b) E4 = Eijk, Eijk | i, j, k ∈ Z (a ﬁnite set), where i j k Eijk = hz, z1, A, [[X B, X B, X B]]i and ′ b i j k Eijk = hz, z1, A, [[X YB,X YB,X YB]]i.

i j k ′ i j k (c) xC(Eijk) = [[(−I) , (−I) , (−I)b ]] and xC(Eijk) = [[(−I) , (−I) , (−I) ]]·A. (d) All of the subgroups E′ have type II. The subgroup E has type I if ijk ijk b and only if i ≡ j (mod 2), and lies in C (is conjugate to E∗ ) if and only if i ≡ j ≡ k (mod 2). The subgroups E000 , E001 , and E100 thus represent the three conjugacy classes of rank four elementary abelian subgroups of n Spin7(q ) (and E∗ = E000 ). n ′ ′′ U (e) For any ϕ ∈ Γn ≤ Aut(H(q )) (see Deﬁnition 2.4), if E , E ∈ E4 are ′ ′′ ′ ′′ such that ϕ(E )= E , then ϕ(xC (E )) = xC(E ).

Proof (a) The set (I, 0) , (A, 0) , (B, 0) , (AB, 0) , (0, A) , (0,B) , (0,AB) 0 is a basis of eigenvectors for the action of E∗ on Vn = M2(Fqn ) ⊕ M2 (Fqn ). (Since the matrices A, B, and AB all have order 4 and determinant one, each has as eigenvalues the two distinct fourth roots of unity, and hence they all have trace zero.) Since all of these have determinant one, E∗ has type I by deﬁnition.

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(b) Consider the subgroups ∞ 3 n i j k i j k R0 = ω(C(q ) ) ∩ S(q )= [[X , X , X ]], [[X Y, X Y, X Y ]] i, j, k ∈ Z and

R1 = CS(qn)(hU, Ai)= R0·hBi. U Clearly, each subgroup E ∈E4 is containedb in b n i j k CS(qn)(U)= S0(q )= R0·h[[B ,B ,B ]]i.

All involutions in this subgroup are contained in R1 = R0·h[[B,B,B]]i, and thus E ≤ R1 . Hence E ∩ R0 has rank 3, which implies that E ≥ hz, z1, Ai (the 2–torsion in R0 ). Since all elements of order two in the coset R0·B have the form b [[XiB, XjB, XkB]] or [[XiYB,XjYB,XkYB]] b ′ for some i, j, k, this shows that E must be one of the groups Eijk or Eijk . (Note in particular that E∗ = E000 .)

(c) By Proposition A.9(a), the element xC(E) ∈ E is characterized uniquely −1 qn ∞ by the property that xC(E) = g ψ (g) for some g ∈ Spin7(q ) such that −1 ′ gEg ∈C . We now apply this explicitly to the subgroups Eijk and Eijk . For each i, Y −i(XiB)Y i = Y −2iXiB = B. Hence for each i, j, k, i j k −1 i j k [[Y ,Y ,Y ]] ·Eijk·[[Y ,Y ,Y ]] = E∗ and n ψq ([[Y i,Y j,Y k]]) = [[Y i,Y j,Y k]]·[[(−I)i, (−I)j , (−I)k]]. Hence i j k xC(Eijk) = [[(−I) , (−I) , (−I) ]].

∞ 2 Similarly, if we choose Z ∈ CSL2(q )(A) such that Z = Y , then for each i, (Y iZ)−1(XiYB)(Y iZ)= B. Hence for each i, j, k, i j k −1 ′ i j k [[Y Z,Y Z,Y Z]] ·Eijk·[[Y Z,Y Z,Y Z]] = E∗. n Since ψq (Z)= ±ZA, n ψq ([[Y iZ,Y jZ,Y kZ]]) = [[Y iZ,Y jZ,Y kZ]]·[[(−I)iA, (−I)jA, (−I)kA]], and hence ′ i j k xC (Eijk) = [[(−I) A, (−I) A, (−I) A]].

Geometry & Topology, Volume 6 (2002) 936 Ran Levi and Bob Oliver

(d) This now follows immediately from point (c) and Proposition A.9(b,c).

n (e) By Deﬁnition 2.4, Γn is generated by Inn(H(q )) and the permutations ∞ ∞ 3 of the three factors in H(q ) =∼ (SL2(q ) )/{±(I,I,I)}. If ϕ ∈ Γn is a U permutation automorphism, then it permutes the elements of E4 , and preserves n ′ ′′ the elements xC(−) by the formulas in (c). If ϕ ∈ Inn(H(q )) and ϕ(E )= E ′ ′′ U ′ ′′ for E , E ∈ E4 , then ϕ(xC (E )) = xC(E ) by deﬁnition of xC(−); and so the same property holds for all elements of Γn .

Following the notation introduced in Section 1, Hom n (P,Q) (for P,Q ≤ Spin7(q ) S(qn)) denotes the set of homomorphisms from P to Q induced by conjugation n n n by some element of Spin7(q ). Also, if P,Q ≤ S(q ) ∩ H(q ), HomΓn (P,Q) denotes the set of homomorphisms induced by restriction of an element of Γn . n n n Let Fn = FSol(q ) be the fusion system over S(q ) generated by Spin7(q ) n and Γn . In other words, for each P,Q ≤ S(q ), HomFn (P,Q) is the set of all composites

ϕ1 ϕ2 ϕk P = P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−→ Pk = Q, n n where Pi ≤ S(q ) for all i, and each ϕi lies in HomSpin7(q )(Pi−1, Pi) or (if n Pi−1, Pi ≤ H(q )) HomΓn (Pi−1, Pi). This clearly deﬁnes a fusion system over S(qn).

Proposition 2.9 Fix n ≥ 1. Let E ≤ S(qn) be an elementary abelian subgroup of rank 3 which contains U , and such that

n n CS(q )(E) ∈ Syl2(CSpin7(q )(E)). Then

{ϕ ∈ Aut (C n (E)) | ϕ(z)= z} = Aut n (C n (E)). (1) Fn S(q ) Spin7(q ) S(q )

Proof Set n n Spin = Spin7(q ), S = S(q ), Γ=Γn, and F = Fn for short. Consider the subgroups

n def ∞ 3 n def R0 = R0(q ) = ω(C(q ) ) ∩ S and R1 = R1(q ) = CS(hU, Ai)= hR0, Bi.

Here, R0 is generated by elements of the form [[X1, X2, X3]], where either Xi ∈ n 2n qn b nb C(q ), or X1 = X2 = X3 = X ∈ C(q ) and ψ (X) = −X . Also, C(q ) ∈ k k n Syl2(CSL2(q )(A)) is cyclic of order 2 ≥ 4, where 2 is the largest power which divides qn ± 1; and C(q2n) is cyclic of order 2k+1 . So ∼ 3 R0 = (C2k ) and R1 = R0 ⋊ hBi,

b Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 937

−1 where B = [[B,B,B]] has order 2 and acts on R0 via (g 7→ g ). Note that hU, Ai = h[[±I, ±I, ±I]], [[A, A, A]]i =∼ C3 b 2 is the 2–torsion subgroup of R . b 0 We claim that 3 R0 is the only subgroup of S isomorphic to (C2k ) . (2) ′ 3 ′ ∼ 3 To see this, let R ≤ S be any subgroup isomorphic to (C2k ) , and let E = C2 be its 2–torsion subgroup. Recall that for any 2–group P , the Frattini subgroup Fr(P ) is the subgroup generated by commutators and squares in P . Thus ′ ′ E ≤ Fr(R ) ≤ Fr(S) ≤ hR0, [[B,B,I]]i (note that [[B,B,I]] = (τ·[[B,I,I]])2 ). Any elementary abelian subgroup of ∼ 3 rank 4 in Fr(S) would have to contain hU, Ai (the 2–torsion in R0 = C2k ), and this is impossible since no element of the coset R0·[[B,B,I]] commutes with A. Thus, rk(Fr(S)) = 3. Hence U ≤ E′ , sinceb otherwise hU, E′i would be an elementary abelian subgroup of Fr(S) of rank ≥ 4. This in turn implies that ′ ′ ′ Rb ≤ CS(U), and hence that E ≤ Fr(CS (U)) ≤ R0 . Thus E = hU, Ai (the ′ 2–torsion in R0 again). Hence R ≤ CS(hU, Ai)= hR0, Bi, and it follows that ′ R = R0 . This ﬁnishes the proof of (2). b b b ∞ Choose generators x1,x2,x3 ∈ R0 as follows. Fix X ∈ CSL2(q )(A) of order k k+1 2 2n 2 , and Y ∈ CSL2(q )(A) of order 2 such that Y = X . Set x1 = [[I,I,X]], 2k−1 2k−1 x2 = [[X,I,I]], and x3 = [[Y,Y,Y ]]. Thus, x1 = z, x2 = z1 , and 2k−1 (x3) = A. Now let E ≤ S(qn) be an elementary abelian subgroup of rank 3 which con- b tains U , and such that CS(qn)(E) ∈ Syl2(CSpin(E)). In particular, E ≤ R1 = CS(qn)(U). There are two cases to consider: that where E ≤ R0 and that where E R0 .

Case 1: Assume E ≤ R0 . Since R0 is abelian of rank 3, we must have E = hU, Ai, the 2–torsion subgroup of R0 , and CS(E) = R1 . Also, by (2), neither R0 nor R1 is isomorphic to any other subgroup of S ; and hence b AutF (Ri)= AutSpin(Ri), AutΓ(Ri) for i = 0, 1. (4)

By Proposition A.8, AutSpin(E) is the group of all automorphisms of E which n send z to itself. In particular, since H(q ) = CSpin(U), AutH(qn)(E) is the group of all automorphisms of E which are the identity on U . Also, Γ = n Inn(H(q ))·Σ3 , where Σ3 sends A = [[A, A, A]] to itself and permutes the nontrivial elements of U = {[[±I, ±I, ±I]]}. Hence AutΓ(E) is the group of all b b b

Geometry & Topology, Volume 6 (2002) 938 Ran Levi and Bob Oliver

∼ automorphisms which send U to itself. So if we identify Aut(E) = GL3(Z/2) via the basis {z, z1, A}, then

def 1 AutSpin(E)= Tb1 = GL2(Z/2) = (aij) ∈ GL3(Z/2) | a21 = a31 = 0 and def 2 AutΓ(E)= T2 = GL1(Z/2) = (aij) ∈ GL3(Z/2) | a31 = a32 = 0 .

By (2) (and since E is the 2–torsion in R0 ),

NSpin(E)= NSpin(R0) and {γ ∈ Γ | γ(E)= E} = {γ ∈ Γ | γ(R0)= R0}.

Since CSpin(E) = CSpin(R0)·hBi, the only nonidentity element of AutSpin(R0) or of AutΓ(R0) which is the identity on E is conjugation by B, which is −I . Hence restriction from R0 to Eb induces isomorphisms b AutSpin(R0)/{±I} =∼ AutSpin(E) and AutΓ(R0)/{±I} =∼ AutΓ(E). k Upon identifying Aut(R0) =∼ GL3(Z/2 ) via the basis {x1,x2,x3}, these can be regarded as sections k k k ∗ µi : Ti −−−−−→ GL3(Z/2 )/{±I} = SL3(Z/2 ) × {λI | λ ∈ (Z/2 ) }/{±I} k of the natural projection from GL3(Z/2 )/{±I} to GL3(Z/2), which agree on the group T0 = T1 ∩ T2 of upper triangular matrices.

We claim that µ1 and µ2 both map trivially to the second factor. Since this factor is abelian, it suﬃces to show that T0 is generated by [T1, T1] ∩ T0 and [T2, T2] ∩ T0 , and that each Ti is generated by [Ti, Ti] and T0 — and this is easily checked. (Note that T1 =∼ T2 =∼ Σ4 .)

By carrying out the above procedure over the ﬁeld Fq2n , we see that both of k+1 these sections µi can be lifted further to SL3(Z/2 ) (still agreeing on T0 ). So by Lemma A.10, there is a section k µ: GL3(Z/2) −−−−−→ SL3(Z/2 ) which extends both µ1 and µ2 . By (4), AutF (R0) = Im(µ)·h− Ii.

def n We next identify AutF (R1). By Lemma 2.8(a), E∗ = hz, z1, A, Bi≤ Spin7(q ) is a subgroup of rank 4 and type I. So by Proposition A.8, AutSpin(E∗) contains ∼ 4 all automorphisms of E∗ = C2 which send z ∈ Z(Spin) to itself.b b Hence for any x ∈ NSpin(R1), since cx(z)= z, there is x1 ∈ NSpin(E∗) such that cx1 |E = cx|E −1 −1 (ie, xx1 ∈ CSpin(E)) and cx1 (B)= B (ie, [x1, B] = 1). Set x2 = xx1 . Since C (U)= H(qn) ≤ Im(ω), we see that C (E)= K ·hBi, where Spin b b bSpin 0 3 K = ω(C ∞ (A) ) ∩ Spin 0 SL2(q ) b

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 939

is abelian, R0 ∈ Syl2(K0), and B acts on K0 by inversion. Upon replacing x1 −1 by Bx1 and x2 by x2B if necessary, we can assume that x2 ∈ K0 . Then b [x , B]= x ·(Bx B−1)−1 = x2, b b 2 2 2 2 while by the original choice of x,x we have b 1 b b −1 [x2, B] = [xx1 , B] = [x, B] ∈ R0. Thus x2 ∈ R ∈ Syl (K ), and hence x ∈ R ≤ R . Since x = x x was an 2 0 2 0 b b2 0 b 1 2 1 arbitrary element of NSpin(R1), this shows that NSpin(R1) ≤ R1·CSpin(B), and hence that b AutSpin(R1) = Inn(R1)·{ϕ ∈ AutSpin(R1) | ϕ(B)= B}. (5)

Since AutΓ(R1) is generated by its intersection with AutSpinb (R1)b and the group ∞ Σ3 which permutes the three factors in H(q ) (and since the elements of Σ3 all ﬁx B), we also have b b Aut (R ) = Inn(R )·{ϕ ∈ Aut (R ) | ϕ(B)= B}. b Γ 1 1 Γ 1 Together with (4) and (5), this shows that Aut (R ) is generated by Inn(R ) F 1 b b 1 together with certain automorphisms of R1 = R0·hBi which send B to itself. In other words, b b AutF (R1) = Inn(R1)· ϕ ∈ Aut(R1) ϕ(B)= B, ϕ|R0 ∈ AutF (R0)

= Inn(R1)·ϕ ∈ Aut(R1) ϕ(B)= B, ϕ|R0 ∈ µ(GL3(Z/ 2)) . b b Thus b b ϕ ∈ AutF (R1) ϕ(z)= z

= Inn(R1)· ϕ ∈ Aut(R 1) ϕ(B)= B, ϕ|R0 ∈ µ(T1) = AutSpin(R0)

= AutSpin( R1), b b the last equality by (5); and (1) now follows.

Case 2: Now assume that E R0 . By assumption, U ≤ E (hence E ≤ CS(E) ≤ CS(U)), and CS(E) is a Sylow subgroup of CSpin(E). Since CS(E) is not isomorphic to R1 = CS(hz, z1, Ai) (by (2)), this shows that E is not Spin–conjugate to hz, z1, Ai. By Proposition A.8, Spin contains exactly two conjugacy classes of rank 3 subgroups containingb z, and thus E must have type II. Hence by Proposition A.8(d),b CS(E) is elementary abelian of rank 4, and also has type II. n ∼ 4 Let C be the Spin7(q )–conjugacy class of the subgroup E∗ = hU, A, Bi = C2 , which by Lemma 2.8(a) has type I. Let E′ be the set of all subgroups of S which b b

Geometry & Topology, Volume 6 (2002) 940 Ran Levi and Bob Oliver are elementary abelian of rank 4, contain U , and are not in C . By Lemma ′ ′′ ′ ′ ′′ def ′ ′ 2.8(e), for any ϕ ∈ IsoΓ(E , E ) and any E ∈ E , E = ϕ(E ) ∈ E , and ϕ ′ ′′ ′ ′′ sends xC(E ) to xC(E ). The same holds for ϕ ∈ IsoSpin(E , E ) by deﬁnition ′ of the elements xC(−) (Proposition A.9). Since CS(E) ∈E , this shows that all elements of AutF (CS(E)) send the element xC(CS(E)) to itself. By Proposition A.9(c), AutSpin(CS (E)) is the group of automorphisms which are the identity on the rank two subgroup hxC(CS(E)), zi; and (1) now follows.

One more technical result is needed.

Lemma 2.10 Fix n ≥ 1, and let E, E′ ≤ S(qn) be two elementary abelian subgroups of rank three which contain U , and which are Γn –conjugate. Then ′ n E and E are Spin7(q )–conjugate.

∞ Proof By [23, 3.6.3(ii)], −I is the only element of order 2 in SL2(q ). Con- sider the sets n 2 J1 = X ∈ SL2(q ) X = −I and 2n qn 2 J2 = X ∈ SL2(q ) ψ (X)= −X, X = −I . qn qn Here, as usual, ψ is induced by the ﬁeld automorphism (x 7→ x ). All elements in J1 are SL2(q)–conjugate (this follows, for example, from [23, 3.6.23]), and we claim the same is true for elements of J2 . ∗ n 2n qn Let SL2(q ) be the group of all elements X ∈ SL2(q ) such that ψ (X) = n ±X . This is a group which contains SL2(q ) with index 2. Let k be such that n k n the Sylow 2–subgroups of SL2(q ) have order 2 ; then k ≥ 3 since |SL2(q )| = n 2n ∗ n k+1 q (q − 1). Any S ∈ Syl2(SL2(q )) is quaternion of order 2 ≥ 16 (see [15, n k Theorem 2.8.3]) and its intersection with SL2(q ) is quaternion of order 2 , so all elements in S ∩J2 are S –conjugate. It follows that all elements of J2 ∗ n ′ ′ −1 ∗ n are SL2(q )–conjugate. If X, X ∈ J2 and X = gXg for g ∈ SL2(q ), n n ′ then either g ∈ SL2(q ) or gX ∈ SL2(q ), and in either case X and X are n conjugate by an element of SL2(q ). By Proposition 2.5(a),

′ n def ∞ 3 n E, E ≤ C n (U)= H(q ) = ω(SL (q ) ) ∩ Spin (q ). Spin7(q ) 2 7 ′ ′ ′ ′ Thus E = hz, z1, [[X1, X2, X3]]i and E = hz, z1, [[X1, X2, X3]]i, where the Xi ′ ′ are all in J1 or all in J2 , and similarly for the Xi . Also, since E and E are Γn –conjugate (and each element of Γn leaves U = hz, z1i invariant), the Xi and ′ n Xi must all be in the same set J1 or J2 . Hence they are all SL2(q )–conjugate, ′ n and so E and E are Spin7(q )–conjugate.

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 941

We are now ready to show that the fusion systems Fn are saturated, and satisfy the conditions listed in Theorem 2.1.

Proposition 2.11 For a fixed odd prime power q, let S(qn) ≤ S(q∞) ≤ ∞ ∞ Spin7(q ) be as defined above. Let z ∈ Z(Spin7(q )) be the central element n of order 2. Then for each n, Fn = FSol(q ) is saturated as a fusion system over S(qn), and satisfies the following conditions:

(a) For all P,Q ≤ S(qn) which contain z, if α ∈ Hom(P,Q) is such that

n α(z)= z, then α ∈ HomFn (P,Q) if and only if α ∈ HomSpin7(q )(P,Q). n n (b) CFn (z)= FS(qn)(Spin7(q )) as fusion systems over S(q ). n (c) All involutions of S(q ) are Fn –conjugate.

Furthermore, Fm ⊆ Fn for m|n. The union of the Fn is thus a category ∞ ∞ FSol(q ) whose objects are the ﬁnite subgroups of S(q ).

n n Proof We apply Proposition 1.2, where p = 2, G = Spin7(q ), S = S(q ), n Z = hzi = Z(G); and U and CG(U)= H(q ) are as deﬁned above. Also, Γ = n Γn ≤ Aut(H(q )). Condition (a) in Proposition 1.2 (all noncentral involutions in G are conjugate) holds since all subgroups in E2 are conjugate (Proposition A.8), and condition (b) holds by deﬁnition of Γ. Condition (c) holds since

n n {γ ∈ Γ | γ(z)= z} = Inn(H(q ))·hcτ i = AutNG(U)(H(q )) n by deﬁnition, since H(q )= CG(U), and by Proposition 2.5(b). Condition (d) was shown in Proposition 2.9, and condition (e) in Lemma 2.10. So by Propo- n sition 1.2, Fn is a saturated fusion system, and CFn (Z)= FS(qn)(Spin7(q )). The last statement is clear.

3 Linking systems and their automorphisms

We next show the existence and uniqueness of centric linking systems associated to the FSol(q), and also construct certain automorphisms of these categories q n analogous to the automorphisms ψ of the group Spin7(q ). One more technical lemma about elementary abelian subgroups, this time about their F –conjugacy classes, is ﬁrst needed.

Geometry & Topology, Volume 6 (2002) 942 Ran Levi and Bob Oliver

Lemma 3.1 Set F = FSol(q). For each r ≤ 3, there is a unique F –conjugacy class of elementary abelian subgroups E ≤ S(q) of rank r. There are two F –conjugacy classes of rank four elementary abelian subgroups E ≤ S(q): one is the set C of subgroups Spin7(q)–conjugate to E∗ = hz, z1, A, Bi, while the other contains the other conjugacy class of type I subgroups as well as all type II subgroups. Furthermore, AutF (E) = Aut(E) for all elementaryb b abelian subgroups E ≤ S(q) except when E has rank four and is not F –conjugate to E∗ , in which case

AutF (E)= {α ∈ Aut(E) | α(xC (E)) = xC(E)}.

Proof By Lemma 2.8(d), the three subgroups

E∗ = hz,z1, A, [[B,B,B]]i, E001 = hz,z1, A, [[B,B,XB]]i, E100 = hz,z1, A, [[XB,B,B]]i

(where X isb a generator of C(q)) representb the three Spin7(q)–conjugacyb classes of rank four subgroups. Clearly, E100 and E001 are Γ1 –conjugate, hence F – conjugate; and by Lemma 2.8(e), neither is Γ1 –conjugate to E∗ . This proves that there are exactly two F –conjugacy classes of such subgroups.

Since E∗ and E001 both are of type I in Spin7(q), their Spin7(q)–automorphism groups contain all automorphisms which ﬁx z (see Proposition A.8). By Lemma 2.8(e), z is ﬁxed by all Γ–automorphisms of E001 , and so AutF (E001) is the group of all automorphisms of E001 which send z = xC(E001) to itself. On the other hand, E∗ contains automorphisms (induced by permuting the three coor- dinates of H ) which permute the three elements z, z1, zz1 ; and these together with AutSpin(E∗) generate Aut(E∗). It remains to deal with the subgroups of smaller rank. By Proposition A.8 again, there is just one Spin7(q)–conjugacy class of elementary abelian subgroups of rank one or two. There are two conjugacy classes of rank three subgroups, those of type I and those of type II. Since E100 is of type II and E001 of type I, all rank three subgroups of E001 have type I, while some of the rank three subgroups of E100 have type II. Since E001 is F –conjugate to E100 , this shows that some subgroup of rank three and type II is F –conjugate to a subgroup of type I, and hence all rank three subgroups are conjugate to each other. Finally, AutF (E) = Aut(E) whenver rk(E) ≤ 3 since any such group is F –conjugate to a subgroup of E∗ (and we have just seen that AutF (E∗) = Aut(E∗)).

To simplify the notation, we now deﬁne

n def n FSpin(q ) = FS(qn)(Spin7(q ))

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 943

n for all 1 ≤ n ≤ ∞: the fusion system of the group Spin7(q ) at the Sylow n n subgroup S(q ). By construction, this is a subcategory of FSol(q ). We write n n n n OSol(q )= O(FSol(q )) and OSpin(q )= O(FSpin(q )) for the corresponding orbit categories: both of these have as objects the subgroups of S(qn), and have as morphism sets

n n MorOSol(q )(P,Q) = HomFSol(q )(P,Q)/ Inn(Q) ⊆ Rep(P,Q) and

n n MorOSpin(q )(P,Q) = HomFSpin(q )(P,Q)/ Inn(Q) . c n n c n n Let OSol(q ) ⊆ OSol(q ) and OSpin(q ) ⊆ OSpin(q ) be the centric orbit cate- n n gories; ie, the full subcategories whose objects are the FSol(q )– or FSpin(q )– centric subgroups of S(qn). (We will see shortly that these in fact have the same objects.) The obstructions to the existence and uniqueness of linking systems associated n to the fusion systems FSol(q ), and to the existence and uniqueness of certain automorphisms of those linking systems, lie in certain groups which were iden- tiﬁed in [6] and [5]. It is these groups which are shown to vanish in the next lemma.

Lemma 3.2 Fix a prime power q, and let c c ZSol(q): OSol(q) −−−−→ Ab and ZSpin(q): OSpin(q) −−−−→ Ab be the functors Z(P )= Z(P ). Then for all i ≥ 0, i i lim (ZSol(q)) = 0 = lim (ZSpin(q)). ←−c c←− OSol(q) OSpin(q)

Proof Set F = FSol(q) for short. Let P1,...,Pk be F –conjugacy class representatives for all F –centric subgroups Pi ≤ S(q), arranged such that |Pi| ≤ |Pj | for i ≤ j . For each i, let Zi ⊆ ZSol(q) be the subfunctor deﬁned by setting Zi(P ) = ZSol(q)(P ) if P is conjugate to Pj for some j ≤ i and Zi(P ) = 0 otherwise. We thus have a ﬁltration

0= Z0 ⊆Z1 ⊆···⊆Zk = ZSol(q) of ZSol(q) by subfunctors, with the property that for each i, the quotient functor Zi/Zi−1 vanishes except on the conjugacy class of Pi (and such that (Zi/Zi−1)(Pi)= ZSol(q)(Pi)). By [6, Proposition 3.2], ∗ ∼ ∗ ←−lim (Zi/Zi−1) = Λ (OutF (Pi); Z(Pi))

Geometry & Topology, Volume 6 (2002) 944 Ran Levi and Bob Oliver for each i. Here, Λ∗(Γ; M) are certain graded groups, defined in [16, section 5] for all finite groups Γ and all finite Z(p)[Γ]–modules M . We will show that ∗ Λ (OutF (Pi); Z(Pi)) = 0 except when Pi = S(q) or S0(q) (see Definition 2.6).

Fix an F –centric subgroup P ≤ S(q). For each j ≥ 1, let Ωj(Z(P )) = {g ∈ 2j Z(P ) | g = 1}, and set E = Ω1(Z(P )) — the 2–torsion in the center of P . 2j For each j ≥ 1, let Ωj(Z(P )) = {g ∈ Z(P ) | g = 1}, and set E = Ω1(Z(P )) — the 2–torsion in the center of P . We can assume E is fully centralized in F (otherwise replace P and E by appropriate subgroups in the same F –conjugacy classes).

def Assume ﬁrst that Q = CS(q)(E) P , and hence that NQ(P ) P . Then any x ∈ NQ(P )rP centralizes E = Ω1(Z(P )). Hence for each j , x acts triv- j−1 ially on Ωj(Z(P ))/Ωj−1(Z(P )), since multiplication by p sends this group NQ(P )/P –linearly and monomorphically to E . Since cx is a nontrivial element of OutF (P ) of p–power order, ∗ Λ (OutF (P ); Ωj(Z(P ))/Ωj−1(Z(P ))) = 0 ∗ for all j ≥ 1 by [16, Proposition 5.5], and thus Λ (OutF (P ); Z(P )) = 0.

Now assume that P = CS(q)(E) = P , the centralizer in S(q) of a fully F – centralized elementary abelian subgroup. Since there is a unique conjugacy class of elementary abelian subgroup of any rank ≤ 3, CS(q)(E) always contains 4 4 a subgroup C2 , and hence P contains a subgroup C2 which is self centralizing by Proposition A.8(a). This shows that Z(P ) is elementary abelian, and hence that Z(P )= E . We can assume P is fully normalized in F , so

AutS(q)(P ) ∈ Syl2(AutF (P )) by condition (I) in the deﬁnition of a saturated fusion system. Since P = CS(q)(E) (and E = Z(P )), this shows that

Ker OutF (P ) −−−→ AutF (E) has odd order. Also, since E is fully centralized, any F –automorphism of E extends to an F –automorphism of P = CS(q)(E), and thus this restriction map between automorphism groups is onto. By [16, Proposition 6.1(i,iii)], it now follows that i i Λ (OutF (P ); Z(P )) =∼ Λ (AutF (E); E). (1)

By Lemma 3.1, AutF (E) = Aut(E), except when E lies in one certain F – ∼ 4 conjugacy class of subgroups E = C2 ; and in this case P = E and AutF (E) is

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 945

the group of automorphisms ﬁxing the element xC(E). In this last (exceptional) case, O2(AutF (E)) 6= 1 (the subgroup of elements which are the identity on E/hxC(E)i), so ∗ ∗ Λ (OutF (P ); Z(P )) = Λ (AutF (E); E) = 0 (2) by [16, Proposition 6.1(ii)]. Otherwise, when AutF (E) = Aut(E), by [16, Proposition 6.3] we have Z/2 if rk(E) = 2, i = 1 i ∼ Λ (AutF (E); E) = Z/2 if rk(E) = 1, i = 0 (3) 0 otherwise. ∗ By points (1), (2), and (3), the groups Λ (OutF (P ); Z(P )) vanish except in the two cases E = hzi or E = U , and these correspond to P = S(q) or P = NS(q)(U)= S0(q).

We can assume that Pk = S(q) and Pk−1 = S0(q). We have now shown that ∗ ←−lim (Zk−2) = 0, and thus that ZSol(q) has the same higher limits as Zk/Zk−2 . j Hence←− lim (ZSol(q)) = 0 for all j ≥ 2, and there is an exact sequence 0 0 1 0 −−−→ ←−lim (ZSol(q)) −−−−→ ←−lim (Zk/Zk−1) −−−−→ ←−lim (Zk−1/Zk−2) =∼Z/2 =∼Z/2 1 −−−−→ ←−lim (ZSol(q)) −−−→ 0. 0 1 One easily checks that lim←− (ZSol(q)) = 0, and hence we also get lim←− (ZSol(q)) = 0. i The proof that←− lim (ZSpin(q)) = 0 for all i ≥ 1 is similar, but simpler. If F = FSpin(q), then for any F –centric subgroup P S(q), there is an element x ∈ NS(P )rP such that [x, P ] = hzi, and cx is a nontrivial element of O2(OutF (P )). Thus ∗ Λ (OutF (P ); Z(P )) = 0 for all such P by [16, Proposition 6.1(ii)] again.

We are now ready to construct classifying spaces BSol(q) for these fusion systems FSol(q). The following proposition ﬁnishes the proof of Theorem 2.1, and also contains additional information about the spaces BSol(q). c n c n To simplify notation, we write LSpin(q ) = LS(qn)(Spin7(q )) (n ≥ 1) to de- n note the centric linking system for the group Spin7(q ). The ﬁeld automor- q n n phism (x 7→ x ) induces an automorphism of Spin7(q ) which sends S(q ) to q q q itself; and this in turn induces automorphisms ψF = ψF (Sol), ψF (Spin), and q n n ψL(Spin) of the fusion systems FSol(q ) ⊇ FSpin(q ) and of the linking system c n LSpin(q ).

Geometry & Topology, Volume 6 (2002) 946 Ran Levi and Bob Oliver

Proposition 3.3 Fix an odd prime q, and n ≥ 1. Let S = S(qn) ∈ n n Syl2(Spin7(q )) be as deﬁned above. Let z ∈ Z(Spin7(q )) be the central element of order 2. Then there is a centric linking system c n π n L = LSol(q ) −−−−−→FSol(q ) def n associated to the saturated fusion system F = FSol(q ) over S , which has the following additional properties.

n (a) A subgroup P ≤ S is F –centric if and only if it is FSpin(q )–centric. c n c n (b) L (q ) contains L (q ) as a subcategory, in such a way that π| c n Sol Spin LSpin(q ) c n is the usual projection to FSpin(q ), and that the distinguished monomorphisms δP P −−−→ AutL(P ) c n c n for L = LSol(q ) are the same as those for LSpin(q ). c n c n (c) Each automorphism of LSpin(q ) which covers the identity on FSpin(q ) c n extends to an automorphism of LSol(q ) which covers the identity on c n FSol(q ). Furthermore, such an extension is unique up to composition with the functor c n c n Cz : LSol(q ) −−−−−→LSol(q )

which is the identity on objects and sends α ∈ Mor c n (P,Q) to z ◦ α ◦ LSol(q ) z−1 (“conjugation by z”). q c n b (d) There is a unique automorphism ψL ∈ Aut(LSol(q )) which covers the b n q automorphism of FSol(q ) induced by the ﬁeld automorphism (x 7→ x ), c n which extends the automorphism of LSpin(q ) induced by the ﬁeld auto- −1 morphism, and which is the identity on π (FSol(q)).

n Proof By Proposition 2.11, F = FSol(q ) is a saturated fusion system over n n n S = S(q ) ∈ Syl2(Spin7(q )), with the property that CF (z)= FSpin(q ). Point (a) follows as a special case of [6, Proposition 2.5(a)]. i n Since←− lim (ZSol(q )) = 0 for i = 2, 3 by Lemma 3.2, there is by [6, Propo- c n OSol(q ) c n sition 3.1] a centric linking system L = LSol(q ) associated to F , which is unique up to isomorphism (an isomorphism which commutes with the projec- n tion to FSol(q ) and with the distinguished monomorphisms). Furthermore, −1 n n π (FSpin(q )) is a linking system associated to FSpin(q ), such a linking sys- 2 n tem is unique up to isomorphism since lim←− (ZSpin(q )) = 0 (Lemma 3.2 again), and this proves (b).

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(c) By [5, Theorem 6.2] (more precisely, by the same proof as that used in i n [5]), the vanishing of lim (ZSol(q )) for i = 1, 2 (Lemma 3.2) shows that each ←− n automorphism of F = FSol(q ) lifts to an automorphism of L, which is unique up to a natural isomorphism of functors; and any such natural isomorphism sends each object P ≤ S to a isomorphism g for some g ∈ Z(P ). Similarly, i n the vanishing of lim (ZSpin(q )) for i = 1, 2 shows that each automorphism of n ←− c n FSpin(q ) lifts to an automorphism of LSpin(bq ), also unique up to a natural c n c n isomorphism of functors. Since LSol(q ) and LSpin(q ) have the same objects by c n (a), this shows that each automorphism of LSpin(q ) which covers the identity c n c n on FSpin(q ) extends to a unique automorphism of LSol(q ) which covers the n identity on FSol(q ). c n It remains to show, for any Φ ∈ Aut(LSol(q )) which covers the identity on c n F (q ) and such that Φ| c n = Id, that Φ is the identity or conjugation Sol LSpin(q ) by z. We have already noted that Φ must be naturally isomorphic to the c n identity; ie, that there are elements γ(P ) ∈ Z(P ), for all P in LSol(q ), such that −1 Φ(α)= γ(Q) ◦ α ◦ γ(P ) for all α ∈ Mor c n (P,Q), all P,Q. LSol(q ) c n Since Φ is the identity on LSpin(q ), the only possibilities are γ(P ) = 1 for all P (hence Φ = Id), or γ(P )= z for all P (hence Φ is conjugation by z). q n (d) Now consider the automorphism ψF ∈ Aut(FSol(q )) induced by the field q automorphism (x 7→ x ) of Fqn . We have just seen that this lifts to an au- q c n tomorphism ψL of LSol(q ), which is unique up to natural isomorphism of q c n q functors. The restriction of ψL to LSpin(q ), and the automorphism ψL(Spin) c n of LSpin(q ) induced directly by the field automorphism, are two liftings of q n ψF |FSpin(q ) , and hence differ by a natural isomorphism of functors which ex- c n tends to a natural isomorphism of functors on LSol(q ). Upon composing with q this natural isomorphism, we can thus assume that ψL does restrict to the c n automorphism of LSpin(q ) induced by the field automorphism. q Now consider the action of ψL on AutL(S0(q)), which by assumption is the identity on Aut c (S (q)), and in particular on δ(S (q)) itself. Thus, with LSpin(q) 0 0 respect to the extension

1 −−−→ S0(q) −−−−→ AutL(S0(q)) −−−−→ Σ3 −−−→ 1, q ψL is the identity on the kernel and on the quotient, and hence is described by a cocycle 1 1 2 η ∈ Z (Σ3; Z(S0(q))) =∼ Z (Σ3; (Z/2) ). 1 2 q Since H (Σ3; (Z/2) )=0, η must be a coboundary, and thus the action of ψL on AutL(S0(q)) is conjugation by an element of Z(S0(q)). Since it is the identity

Geometry & Topology, Volume 6 (2002) 948 Ran Levi and Bob Oliver

on Aut c (S (q)), it must be conjugation by 1 or z. If it is conjugation by LSpin(q) 0 q z, then we can replace ψL (on the whole category L) by its composite with z; ie, by its composite with the functor which is the identity on objects and sends α ∈ MorL(P,Q) to z ◦ α ◦ z. q In this way, we can assume that ψ is the identity on AutL(S0(q)). By con- b b L struction, every morphism in FSol(q) is a composite of morphisms in FSpin(q) q and restrictions of automorphisms in FSol(q) of S0(q). Since ψL is the identity −1 −1 on π (FSpin(q)), this shows that it is the identity on π (FSol(q)). q ′ It remains to check the uniqueness of ψL . If ψ is another functor with the ′ −1 q same properties, then by (e), (ψ ) ◦ ψL is either the identity or conjugation by z; and the latter is not possible since conjugation by z is not the identity −1 on π (FSol(q)).

∧ This ﬁnishes the construction of the classifying spaces BSol(q) = |LSol(q)|2 for the fusion systems constructed in Section 2. We end the section with an explanation of why these are not the fusion systems of ﬁnite groups.

Proposition 3.4 For any odd prime power q, there is no ﬁnite group G whose fusion system is isomorphic to that of FSol(q).

∼ Proof Let G be a finite group, fix S ∈ Syl2(G), and assume that S = S(q) ∈ Syl2(Spin7(q)), and that the fusion system FS(G) satisfies conditions (a) and (b) in Theorem 2.1. In particular, all involutions in G are conjugate, and the centralizer of any involution z ∈ G has the fusion system of Spin7(q). When q ≡±3 (mod 8), Solomon showed [22, Theorem 3.2] that there is no finite group whose fusion system has these properties. When q ≡±1 (mod 8), he showed (in def the same theorem) that there is no such G such that H = CG(z)/O2′ (CG(z)) is isomorphic to a subgroup of Aut(Spin7(q)) which contains Spin7(q) with odd index. (Here, O2′ (−) means largest odd order normal subgroup.)b

Let G be a ﬁnite group whose fusion system is isomorphic to FSol(q), and again def 2′ set H = CG(z)/O2′ (CG(z)) for some involution z ∈ G. Set H = O (H/hzi): the smallest normal subgroup of H/hzi of odd index. Then H has the fu- ∼ ∼ ′ sionb system of Ω7(q) = Spin7(q)/Z(Spin7(q)). We will show that H =bΩ7(q ) ′ 2′ ∼ ′ for some odd prime power q . It thenb follows that O (H) = Spin7(q ), thus contradicting Solomon’s theorem and proving our claim. b ′ The following “classiﬁcation free” argument for proving that H =∼ Ω7(q ) for some q′ was explained to us by Solomon. We refer to the appendix for general

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± ± results about the groups Spinn (q) and Ωn (q). Fix S ∈ Syl2(H). Thus S is isomorphic to a Sylow 2–subgroup of Ω7(q), and has the same fusion. ′ We first claim that H must be simple. By definition (H = O2 (H/hzi)), H has no proper normal subgroup of odd index, and H has no proper normal subgroup of odd order since any such subgroup would lift to anbodd order normal subgroup of H = CG(z)/O2′ (CG(z)). Hence for any proper normal subgroup N ⊳ H , Q def= N ∩ S is a proper normal subgroup of S , which is strongly closed in S withb respect to H in the sense that no element of Q can be H –conjugate to an element of SrQ. Using Lemma A.4(a), one checks that the group Ω7(q) contains three conjugacy classes of involutions, classified by the dimension of their (−1)–eigenspace. It is not hard to see (by taking products) that any subgroup of S which contains all involutions in one of these conjugacy classes contains all involutions in the other two classes as well. Furthermore, S is generated by the set of all of its involutions, and this shows that there are no proper subgroups which are strongly closed in S with respect to H . Since we have already seen that the intersection with S of any proper normal subgroup of H would have to be such a subgroup, this shows that H is simple. Fix an isomorphism ϕ ′ S −−−−−−−→ S ∈ Syl2(Ω7(q)) =∼ which preserves fusion. Choose x′ ∈ S′ whose (−1)–eigenspace is 4–dimension- ′ al, and such that hx i is fully centralized in FS′ (Ω7(q)). Then ′ ∼ + CO7(q)(x ) = O4 (q) × O3(q) + + by Lemma A.4(c). Since Ω4 (q) ≤ O4 (q) and Ω3(q) ≤ O3(q) both have index 4, ′ + CΩ7(q)(x ) is isomorphic to a subgroup of O4 (q)×O3(q) of index 4, and contains ′ ∼ + ′ a normal subgroup K = Ω4 (q)×Ω3(q) of index 4. Since hx i is fully centralized, def ′ ′ ′ ′ ′ ′ CS (x ) is a Sylow 2–subgroup of CΩ7(q)(x ), and hence S0 = S ∩K is a Sylow 2–subgroup of K′ . −1 ′ ′ Set x = ϕ (x ) ∈ S . Since S =∼ S have the same fusion in H and Ω7(q), ∼ ′ ′ ′ CS(x) = CS (x ) have the same fusion in CH (x) and CΩ7(q)(x ). Hence ∼ ′ H1(CH (x); Z(2)) = H1(CΩ7(q)(x ); Z(2))

(homology is determined by fusion), both have order 4, and thus CH (x) also has a unique normal subgroup K ⊳ H of index 4. Set S0 = K ∩ S . Thus ϕ(S0)= ′ S0 , and using Alperin’s fusion theorem one can show that this isomorphism is fusion preserving with respect to the inclusions of Sylow subgroups S0 ≤ K ′ ′ and S0 ≤ K .

Geometry & Topology, Volume 6 (2002) 950 Ran Levi and Bob Oliver

Using the isomorphisms of Proposition A.5: + ∼ ∼ Ω4 (q) = SL2(q) ×hxi SL2(q) and Ω3(q) = PSL2(q), ′ ′ ′ ′ ∼ ′ ∼ we can write K = K1 ×hx′i K2 , where K1 = SL2(q) and K2 = SL2(q) × ′ ′ ′ ′ ′ ′ ′ −1 ′ PSL2(q). Set Si = S ∩Ki ∈ Syl2(Ki); thus S0 = S1×hx′iS2 . Set Si = ϕ (Si), so that S0 = S1 ×hxi S2 is normal of index 4 in CS(x). The fusion system of K thus splits as a central product of fusion systems, one of which is isomorphic to the fusion system of SL2(q). We now apply a theorem of Goldschmidt, which says very roughly that under these conditions, the group K also splits as a central product. To make this more precise, let Ki be the normal closure of Si in K ⊳ CH (x). By [14, Corollary A2], since S1 and S2 are strongly closed in S0 with respect to K ,

[K1, K2] ≤ hxi·O2′ (K).

Using this, it is not hard to check that Si ∈ Syl2(Ki). Thus K1 has same fusion as SL2(q) and is subnormal in CH (x) (K1 ⊳ K ⊳ CH (x)), and an argument similar to that used above to prove the simplicity of H shows that K1/(hxi·O2′ (K1)) is simple. Hence K1 is a 2–component of CH (x) in the sense described by Aschbacher in [1]. By [1, Corollary III], this implies that H must be isomorphic to a Chevalley group of odd characteristic, or to M11 . It is now straightforward to check that among these groups, the only possibility is that ′ ′ H =∼ Ω7(q ) for some odd prime power q .

4 Relation with the Dwyer-Wilkerson space

We now want to examine the relation between the spaces BSol(q) which we have just constructed, and the space BDI(4) constructed by Dwyer and Wilkerson in [9]. Recall that this is a 2–complete space characterized by the property that its cohomology is the Dickson algebra in four variables over F2 ; ie, the GL4(2) ring of invariants F2[x1,x2,x3,x4] . We show, for any odd prime power q, that BDI(4) is homotopy equivalent to the 2–completion of the union of the spaces BSol(qn), and that BSol(q) is homotopy equivalent to the homotopy fixed point set of an Adams map from BDI(4) to itself. c ∞ We would like to define an infinite “linking system” LSol(q ) as the union of the c n ∞ c ∞ ∧ finite categories LSol(q ), and then set BSol(q )= |LSol(q )|2 . The difficulty with this approach is that a subgroup which is centric in the fusion system m n FSol(q ) need not be centric in a larger fusion system FSol(q ) (for m|n). To get cc n c n around this problem, we define LSol(q ) ⊆ LSol(q ) to be the full subcategory

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n ∞ whose objects are those subgroups of S(q ) which are FSol(q )–centric; or k cc n equivalently FSol(q )–centric for all k ∈ nZ. Similarly, we define LSpin(q ) to c n n be the full subcategory of LSpin(q ) whose objects are those subgroups of S(q ) ∞ c ∞ c ∞ which are FSpin(q )–centric. We can then define LSol(q ) and LSpin(q ) to be the unions of these categories. cc n ∧ For these definitions to be useful, we must first show that |LSol(q )|2 has the c n ∧ same homotopy type as |LSol(q )|2 . This is done in the following lemma. Lemma 4.1 For any odd prime power q and any n ≥ 1, the inclusions cc n ∧ c n ∧ cc n ∧ c n ∧ |LSol(q )|2 ⊆ |LSol(q )|2 and |LSpin(q )|2 ⊆ |LSpin(q )|2 are homotopy equivalences.

Proof It clearly suﬃces to show this when n = 1. Recall, for a fusion system F over a p–group S , that a subgroup P ≤ S is F –radical if OutF (P ) is p–reduced; ie, if Op(OutF (P )) = 1. We will show that ∞ all FSol(q)–centric FSol(q)–radical subgroups of S(q) are FSol(q )–centric (1) and similarly ∞ all FSpin(q)–centric FSpin(q)–radical subgroups of S(q) are FSpin(q )–centric. (2) c In other words, (1) says that for each P ≤ S(q) which is an object of LSol(q) cc but not of LSol(q), O2 OutFSol(q)(P ) 6= 1. By [16, Proposition 6.1(ii)], this implies that ∗ ∗ Λ (OutFSol(q)(P ); H (BP ; F2)) = 0. Hence by [6, Propositions 3.2 and 2.2] (and the spectral sequence for a homotopy cc c colimit), the inclusion LSol(q) ⊆LSol(q) induces an isomorphism ∗ c =∼ ∗ cc H |LSol(q)|; F2 −−−−−−→ H |LSol(q)|; F2 , cc ∧ c ∧ cc ∧ c ∧ and thus |LSol(q)|2 ≃ |LSol(q)|2 . The proof that |LSpin(q)|2 ≃ |LSpin(q)|2 is similar, using (2). Point (2) is shown in Proposition A.12, so it remains only to prove (1). Set k F = FSol(q), and set Fk = FSol(q ) for all 1 ≤ k ≤∞. Let E ≤ Z(P ) be the 2–torsion in the center of P , so that P ≤ CS(q)(E). Set hzi if rk(E) = 1

′ hz, z1i if rk(E) = 2 E =  hz, z1, Ai if rk(E) = 3  E if rk(E) = 4  b   Geometry & Topology, Volume 6 (2002) 952 Ran Levi and Bob Oliver in the notation of Definition 2.6. In all cases, E is F –conjugate to E′ by ′ Lemma 3.1. We claim that E is fully centralized in Fk for all k < ∞. This is clear when rk(E′)=1(E′ = Z(S(qk))), follows from Proposition 2.5(a) when rk(E′) = 2, and from Proposition A.8(a) (all rank 4 subgroups are self centralizing) when rk(E′)=4. If rk(E′) = 3, then by Proposition A.8(d), the k k centralizer in Spin7(q ) (hence in S(q )) of any rank 3 subgroup has an abelian subgroup of index 2; and using this (together with the construction of S(qk) ′ in Definition 2.6), one sees that E is fully centralized in Fk . ′ ′ If E 6= E , choose ϕ ∈ HomF (E,S(q)) such that ϕ(E) = E ; then ϕ extends to ϕ ∈ HomF (CS(q)(E),S(q)) by condition (II) in the definition of a saturated fusion system, and we can replace P by ϕ(P ) and E by ϕ(E). We can thus assume that E is fully centralized in Fk for each k< ∞. So by [6, Proposition

2.5(a)], P is Fk –centric if and only if it is CFk (E)–centric; and this also holds ⊳ when k = ∞. Furthermore, since OutCF (E)(P ) OutF (P ), O2(OutCF (E)(P )) is a normal 2–subgroup of OutF (P ), and thus

O2 OutCF (E)(P ) ≤ O2(OutF (P )).

Hence P is CF (E)–radical if it is F –radical. So it remains to show that all CF (E)–centric CF (E)–radical subgroups of S(q) are also CF∞ (E)–centric. (3) ∞ If rk(E) = 1, then CF (E) = FSpin(q) and CF∞ (E) = FSpin(q ), and (3) follows from (2). If rk(E) = 4, then P = E = CS(q∞)(E) by Proposition A.8(a), so P is F∞ –centric, and the result is clear.

If rk(E) = 3, then by Proposition A.8(d), CF (E) ⊆ CF∞ (E) are the fusion systems of a pair of semidirect products A⋊C2 ≤ A∞⋊C2 , where A ≤ A∞ are abelian and C2 acts on A∞ by inversion. Also, E is the full 2–torsion sub- ∞ group of A∞ , since otherwise rk(A∞) > 3 would imply A∞⋊C2 ≤ Spin7(q ) 5 contains a subgroup C2 (contradicting Proposition A.8). If P ≤ A, then either

OutCF (E)(P ) has order 2, which contradicts the assumption that P is radical; or P is elementary abelian and OutCF (E)(P ) = 1, in which case P ≤ Z(A⋊C2) is not centric. Thus P A; P ∩ A ≥ E contains all 2–torsion in A∞ , and hence P is centric in A∞⋊C2 .

If rk(E) = 2, then by Proposition 2.5(a), CF∞ (E) and CF (E) are the fusion systems of the groups ∞ ∞ 3 H(q ) =∼ SL2(q ) /{±(I,I,I)} (4) and ∞ def 3 H(q)= H(q ) ∩ Spin7(q) ≥ H0(q) = SL2(q) /{±(I,I,I)}.

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If P ≤ S(q) is centric and radical in the fusion system of H(q), then by Lemma 3 A.11(c), its intersection with H0(q) =∼ SL2(q) /{±(I,I,I)} is centric and radical in the fusion system of that group. So by Lemma A.11(a,f),

P ∩ H0(q) =∼ (P1 × P2 × P3)/{±(I,I,I)} (5) for some Pi which are centric and radical in the fusion system of SL2(q). Since the Sylow 2–subgroups of SL2(q) are quaternion [15, Theorem 2.8.3], the Pi ∞ must be nonabelian and quaternion, so each Pi/{±I} is centric in PSL2(q ). 3 Hence P ∩ H0(q) is centric in SL2(q) /{±(I,I,I)} by (5), and so P is centric in H(q∞) by (4).

We would like to be able to regard BSpin7(q) as a subcomplex of BSol(q), but there is no simple natural way to do so. Instead, we set

From now on, when we talk about the inclusion of BSpin7(q) into BSol(q), as long as it need only be well deﬁned up to homotopy, we mean the composite

′ ′ BSpin7(q) ≃ BSpin7(q) ⊆ BSol (q) (for some choice of homotopy equivalence). Similarly, if we talk about the inclusion of BSol(qm) into BSol(qn) (for m|n) where it need only be deﬁned up to homotopy, we mean these spaces identiﬁed with their equivalent subcomplexes BSol′(qm) ⊆ BSol′(qn).

Lemma 4.2 Let q be any odd prime. Then for all n ≥ 1,

∗ n ∗ n C3 H (BSol(q ); F2) → H (BH(q ); F2) (1) ↓ ↓ ∗ n ∗ n H (BSpin7(q ); F2) → H (BH(q ); F2) (with all maps induced by inclusions of groups or spaces) is a pullback square.

Geometry & Topology, Volume 6 (2002) 954 Ran Levi and Bob Oliver

∗ n Proof By [6, Theorem B], H (BSol(q ); F2) is the ring of elements in the cohomology of S(qn) which are stable relative to the fusion. By the construction n n in Section 2, the fusion in Sol(q ) is generated by that in Spin7(q ), together n n with the permutation action of C3 on the subgroup H(q ) ≤ Spin7(q ), and hence (1) is a pullback square.

c ∞ Proposition 4.3 For each odd prime q, there is a category LSol(q ), together with a functor c ∞ ∞ π : LSol(q ) −−−−−−→FSol(q ), such that the following hold:

−1 n ∼ cc n (a) For each n ≥ 1, π (FSol(q )) = LSol(q ). (b) There is a homotopy equivalence

def η BSol(q∞) = |Lc (q∞)|∧ −−−−−−−→ BDI(4) Sol 2 ≃ such that the following square commutes up to homotopy ∞ ′ ∞ ∧ δ(q ) ∞ BSpin7(q )2 → BSol(q )

η0 ≃ η ≃ (1) ↓ ↓ ∧ δ BSpin(7)2 → BDI(4) . b Here, η0 is the homotopy equivalence of [13], induced by some ﬁxed choice ∞ of embedding of the Witt vectors for Fq into C, while δ(q ) is the union cc n ∧ cc n ∧ of the inclusions |LSpin(q )|2 ⊆ |LSol(q )|2 , and δ is the inclusion arising from the construction of BDI(4) in [9]. b q c ∞ Furthermore, there is an automorphism ψL ∈ Aut(LSol(q )) of categories which satisﬁes the conditions:

q cc n (c) the restriction of ψL to each subcategory LSol(q ) is equal to the restric- q c n tion of ψL ∈ Aut(LSol(q )) as defined in Proposition 3.3(d); q q ∞ (d) ψL covers the automorphism ψF of FSol(q ) induced by the field automorphism (x 7→ xq); and q n cc n (e) for each n, (ψL) fixes LSol(q ).

m n Proof By Proposition 2.11, the inclusions Spin7(q ) ≤ Spin7(q ) for all m|n m n induce inclusions of fusion systems FSol(q ) ⊆ FSol(q ). Since the restriction of

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cc n cc m a linking system over FSol(q ) is a linking system over FSol(q ), the uniqueness cc m of linking systems (Proposition 3.3) implies that we get inclusions LSol(q ) ⊆ cc n c ∞ cc n LSol(q ). We define LSol(q ) to be the union of the finite categories LSol(q ). (More precisely, fix a sequence of positive integers n1|n2|n3|··· such that every positive integer divides some ni , and set ∞ c ∞ cc ni LSol(q )= LSol(q ). i[=1 cc n Then by uniqueness again, we can identify LSol(q ) for each n with the appropriate subcategory.)

c ∞ ∞ cc ni Let π : LSol(q ) −−→FSol(q ) be the union of the projections from LSol(q ) to ni ∞ FSol(q ) ⊆ FSol(q ). Condition (a) is clearly satisfied. Also, using Proposition q c ∞ 3.3(d) and Lemma 4.1, we see that there is an automorphism ψL of LSol(q ) which satisfies conditions (c,d,e) above. (Note that by the fusion theorem as c n shown in [6, Theorem A.10], morphisms in LSol(q ) are generated by those cc n between radical subgroups, and hence by those in LSol(q ).) c ∞ ∧ It remains only to show that |LSol(q )|2 ≃ BDI(4), and to show that square (1) commutes. The space BDI(4) is 2–complete by its construction in [9]. By Lemma 4.1, ∗ ∞ ∼ ∗ c n ∗ n H (BSol(q ); F2) = ←−lim H |LSol(q )|; F2 =←− lim H BSol(q ); F2 . n n n n Hence by Lemma 4.2 (and since the inclusions BSpin7(q ) −−→ BSol(q ) commute with the maps induced by inclusions of fields Fqm ⊆ Fqn ), there is a pullback square

∗ ∞ ∗ ∞ C3 H (BSol(q ); F2) → H (BH(q ); F2) (2) ↓ ↓ ∗ ∞ ∗ ∞ H (BSpin7(q ); F2) → H (BH(q ); F2) . Also, by [13, Theorem 1.4], there are maps ∞ ∞ 3 BSpin7(q ) −−−→ BSpin(7) and BH(q ) −−−→ B SU(2) /{±(I,I,I)} which induce isomorphisms of F2 –cohomology, and hence homotopy equivalences after 2–completion. So by Propositions 4.7 and 4.9 (or more directly by the computations in [9, section 3]), the pullback of the above square is the ∗ 4 ring of Dickson invariants in the polynomial algebra H (BC2 ; F2), and thus ∗ isomorphic to H (BDI(4); F2). Point (b), including the commutativity of (1), now follows from the following lemma.

Geometry & Topology, Volume 6 (2002) 956 Ran Levi and Bob Oliver

∗ Lemma 4.4 Let X be a 2–complete space such that H (X; F2) is the Dickson f algebra in 4 variables. Assume further that there is a map BSpin(7) −−→ X ∗ such that H (f| 4 ; F2) is the inclusion of the Dickson invariants in the poly- BC2 ∗ 4 nomial algebra H (BC2 ; F2). Then X ≃ BDI(4). More precisely, there is a homotopy equivalence between these spaces such that the composite

f BSpin(7) −−−−−−→ X ≃ BDI(4) is the inclusion arising from the construction in [9].

Proof In fact, Notbohm [18, Theorem 1.2] has proven that the lemma holds even without the assumption about BSpin(7) (but with the more precise as- ∗ sumption that H (X; F2) is isomorphic as an algebra over the Steenrod algebra to the Dickson algebra). The result as stated above is much more elementary (and also implicit in [9]), so we sketch the proof here.

∗ ∗ Since H (X; F2) is a polynomial algebra, H (ΩX; F2) is isomorphic as a graded vector space to an exterior algebra on the same number of variables, and in particular is finite. Hence X is a 2–compact group. By [11, Theorem 8.1] (the centralizer decomposition for a p–compact group), there is an F2 –homology equivalence hocolim(α) −−−−−−→≃ X. −−−−−→A Here, A is the category of pairs (V, ϕ), where V is a nontrivial elementary ∗ abelian 2–group, and ϕ : BV −−→ X makes H (BV ; F2) into a finitely gen- ∗ erated module over H (X; F2) (see [10, Proposition 9.11]). Morphisms in A are defined by letting MorA((V, ϕ), (V ′, ϕ′)) be the set of monomorphisms V −−→ V ′ of groups which make the obvious triangle commute up to homotopy. Also,

op α: A −−→ Top is the functor α(V, ϕ) = Map(BV,X)ϕ. By [9, Lemma 1.6(1)] and [17, Th´eor`eme 0.4], A is equivalent to the category of elementary abelian 2–groups E with 1 ≤ rk(E) ≤ 4, whose morphisms consist ϕ of all group monomorphisms. Also, if BC2 −−→ X is the restriction of f to any subgroup C2 ≤ Spin(7), then in the notation of Lannes, ∗ ∗ ∼ ∗ TC2 (H (X; F2); ϕ ) = H (BSpin(7); F2) by [9, Lemmas 16.(3), 3.10 and 3.11], and hence

∗ ∗ H (Map(BC2, X)ϕ; F2) =∼ H (BSpin(7); F2)

Geometry & Topology, Volume 6 (2002) Construction of 2–local finite groups 957 by Lannes [17, Théorème 3.2.1]. This shows that

∧ ∧ Map(BC2, X)ϕ 2 ≃ BSpin(7)2 , and thus that the centralizers of other elementary abelian 2–groups are the ∧ same as their centralizers in BSpin(7)2 . In other words, α is equivalent in the homotopy category to the diagram used in [9] to deﬁne BDI(4). By [9, Proposition 7.7] (and the remarks in its proof), this homotopy functor has a unique homotopy lifting to spaces. So by deﬁnition of BDI(4), ∧ X ≃ hocolim(α) 2 ≃ BDI(4). −−−−−→A

q def q ∞ Set Bψ = |ψL|, a self homotopy equivalence of BSol(q ) ≃ BDI(4). By construction, the restriction of Bψq to the maximal torus of BSol(q∞) is the map induced by x 7→ xq , and hence this is an “Adams map” as deﬁned by Notbohm [18]. In fact, by [18, Theorem 3.5], there is an Adams map from BDI(4) to itself, unique up to homotopy, of degree any 2–adic unit.

Following Benson [3], we define BDI4(q) for any odd prime power q to be the homotopy fixed point set of the Z–action on BSol(q∞) ≃ BDI(4) induced by the Adams map Bψq . By “homotopy fixed point set” in this situation, we mean that the following square is a homotopy pullback: ∞ BDI4(q) → BSol(q )

∆ ↓ ↓ (Id,Bψq ) BSol(q∞) → BSol(q∞) × BSol(q∞). The actual pullback of this square is the subspace BSol(q) of elements ﬁxed by q δ0 Bψ , and we thus have a natural map BSol(q) −−→ BDI4(q).

Theorem 4.5 For any odd prime power q, the natural map

δ0 BSol(q) −−−−−−−→ BDI4(q) ≃ is a homotopy equivalence.

Proof Since BDI(4) is simply connected, the square used to deﬁne BDI4(q) remains a homotopy pullback square after 2–completion by [4, II.5.3]. Thus def c ∧ c BDI4(q) is 2–complete. Also, BSol(q) = |LSol(q)|2 is 2–complete since |LSol(q)| is 2–good [6, Proposition 1.12], and hence it suﬃces to prove that the map

Geometry & Topology, Volume 6 (2002) 958 Ran Levi and Bob Oliver

between these spaces is an F2 –cohomology equivalence. By Lemma 4.2, this means showing that the following commutative square is a pullback square:

∗ ∗ C3 H (BDI4(q); F2) → H (BH(q); F2) (1) ↓ ↓ ∗ ∗ H (BSpin7(q); F2) → H (BH(q); F2) . Here, the maps are induced by the composite ′ ∧ BSpin7(q) ≃ BSpin7(q)2 ⊆ BSol(q) −−−−−−→ BDI4(q) and its restriction to BH(q). Also, by Proposition 4.3(b), the following diagram commutes up to homotopy:

incl ∞ η0 BSpin7(q) → BSpin7(q ) → BSpin(7)

δ(q) δ(q∞) δ (2) ↓ ↓ ↓ incl η BSol(q) → BSol(q∞) → BDIb (4)

By [12, Theorem 12.2], together with [13, section 1], for any connected reductive Lie group G and any algebraic epimorphism ψ on G(Fq) with ﬁnite ﬁxed subgroup, there is a homotopy pullback square

ψ ∧ incl ∧ B(G(Fq) )2 → BG(Fq)2

incl↓ ∆↓ (3) ∧ (Id,Bψ) ∧ ∧ BG(Fq)2 → BG(Fq)2 × BG(Fq)2 . 3 We need to apply this when G = Spin7 or G = H = (SL2) /{±(I,I,I)}. In particular, if ψ = ψq is the automorphism induced by the ﬁeld automorphism q ψ ψ def (x 7→ x ), then Spin7(Fq) = Spin7(q) by Lemma A.3, and H(Fq) = H(q) = H(Fq) ∩ Spin7(q). We thus get a description of BSpin7(q) and BH(q) as homotopy pullbacks.

∧ ∧ By [13, Theorem 1.4], BG(Fq)2 ≃ BG(C)2 . Also, we can replace the complex Lie groups Spin7(C) and H(C) by maximal compact subgroups Spin(7) and def H = SU(2)3/{±(I,I,I)}, since these have the same homotopy type.

∗ ∗ If we set R = H (BG(Fq); F2) =∼ H (BG(C); F2), then there are Eilenberg- Moore spectral sequences ∗ R R ∗ ψ E2 = TorR⊗Rop ( , )=⇒ H (B(G(Fq) ); F2);

Geometry & Topology, Volume 6 (2002) Construction of 2–local finite groups 959 where the (R ⊗ Rop)–module structure on R is defined by setting (a ⊗ b)·x = R a·x·Bψ(b). When G = Spin7 or H , then is a polynomial algebra by Proposi- tion 4.7 and the above remarks, and Bψ acts on R via the identity. The above spectral sequence thus satisfies the hypotheses of [20, Theorem II.3.1], and hence collapses. (Alternatively, note that in this case, E2 is generated multiplicatively 0,∗ −1,∗ R ∗ by E2 and E2 by (5) below.) Similarly, when = H (BDI(4); F2), there ∗ is an analogous spectral sequence which converges to H (BDI4(q); F2), and which collapses for the same reason. By the above remarks, these spectral sequences are natural with respect to the inclusions BH(−) ⊆ BSpin7(−), and q (using the naturality of ψ shown in Proposition 3.3(d)) of BSpin7(−) into BSol(−) or BDI(4).

To simplify the notation, we now write

def ∗ def ∗ def ∗ A = H (BDI(4); F2), B = H (BSpin(7); F2), and C = H (H; F2) to denote these cohomology rings. The Frobenius automorphism ψq acts via the identity on each of them. We claim that the square

∗ A A ∗ C C C3 TorA⊗Aop ( , ) → TorC⊗Cop ( , ) (4) ↓ ↓ ∗ B B ∗ C C TorB⊗Bop ( , ) → TorC⊗Cop ( , ) is a pullback square. Once this has been shown, it then follows that in each degree, square (1) has a ﬁnite ﬁltration under which each quotient is a pullback square. Hence (1) itself is a pullback. R R For any commutative F2 –algebra , let ΩR/F2 denote the –module generated by elements dr for r ∈ R with the relations dr =0 if r ∈ F2 , d(r + s)= dr + ds and d(rs)= r·ds + s·dr.

Let Ω∗ denote the ring of Kähler differentials: the exterior algebra (over R/F2 1 R) of ΩR F = Ω . When R is a polynomial algebra, there are natural / 2 R/F2 identifications ∗ ∗ op R R ∼ R R ∼ TorR⊗R ( , ) = HH∗( ; ) = ΩR/F2 . (5) The first isomorphism holds for arbitrary algebras, and is shown, e.g., in [25, Lemma 9.1.3]. The second holds for smooth algebras over a field [25, Theorem 9.4.7] (and polynomial algebras are smooth as shown in [25, section 9.3.1]). In particular, the isomorphisms (5) hold for R = A, B, C, which are shown to be polynomial algebras in Proposition 4.7 below. Thus, square (4) is isomorphic

Geometry & Topology, Volume 6 (2002) 960 Ran Levi and Bob Oliver to the square Ω∗ → Ω∗ C3 A/F2 C/F2 (6) ↓ ↓ Ω∗ → Ω∗ , B/F2 C/F2 which is shown to be a pullback square in Propositions 4.7 and 4.9 below.

It remains to prove that square (6) in the above proof is a pullback square. In what follows, we let Di(x1,...,xn) denote the i-th Dickson invariant in n n−i variables x1,...,xn . This is the (2 −2 )-th symmetric polynomial in the elements (equivalently in the nonzero elements) of the vector space hx ,...,x i . 1 n F2 We refer to [26] for more detail. Note that what he denotes cn,i is what we call Dn−i(x1,...,xn).

Lemma 4.6 For any n, 2 D1(x1,...,xn+1)= (xn+1 + x)+ D1(x1,...,xn)

x∈hx1,...,xniF Y 2 n 2n 2n−i 2 = xn+1 + xn+1 Di(x1,...,xn)+ D1(x1,...,xn) . Xi=1 Proof The ﬁrst equality is shown in [26, Proposition 1.3(b)]; here we prove them both simultaneously. Set V = hx ,...,x i . Since σ (V ) = 0 whenever n 1 n F2 i n 2n − i is not a power of 2 (cf [26, Proposition 1.1]), 2n D1(x1,...,xn+1)= σi(Vn)·σ2n−i(xn+1 + Vn) i=0 X n

= (xn+1 + x)+ Di(x1,...,xn)·σ2n−i (xn+1 + Vn). xY∈Vn Xi=1 n−1 Also, since σi(Vn)=0 for 0

k n−1 k−i 2n−i 0 if 0

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 961

In the following proposition (and throughout the rest of the section), we work with the polynomial ring F2[x, y, z, w], with the natural action of GL4(F2). Let 2 3 GL2(F2), GL1(F2) ≤ GL4(F2) be the subgroups of automorphisms of V def= hx, y, z, wi which leave invariant F2 2 2 the subspaces hx,yi and hx,y,zi, respectively. Also, let GL2′ (F2) ≤ GL2(F2) be the subgroup of automorphisms which are the identity modulo hx,yi. Thus, when described in terms of block matrices (with respect to the given basis {x, y, z, w}), 3 A X 2 B Y 2 B Y GL1(F2)= 0 1 , GL2(F2)= 0 C , and GL2′ (F2)= 0 I , for A ∈ GL3(F2), X a column vector,B,C ∈ GL2(F2), and Y ∈ M2(F2). We need to make more precise the relation between V (or the polynomial ring F2[x, y, z, w]) and the cohomology of Spin(7). To do this, let W ≤ Spin(7) be the inverse image of the elementary abelian subgroup diag(−1, −1, −1, −1, 1, 1, 1), diag(−1, −1, 1, 1, −1, −1, 1), diag(−1, 1, −1, 1, −1, 1, −1) ≤ SO(7). ∼ 4 ′ Thus, W = C2 . Fix a basis {η, η ,ξ,ζ} for W , where ζ ∈ Z(Spin(7)) is the nontrivial element. Identify V = W ∗ in such a way that {x, y, z, w}⊆ V is the dual basis to {η, η′,ξ,ζ}. This gives an identiﬁcation ∗ H (BW ; F2)= F2[x, y, z, w], arranged such that the action of NSpin(7)(W )/W on V = hx, y, z, wi consists of all automorphisms which leave hx,y,zi invariant, and thus can be identiﬁed 3 with the action of GL1(F2). Finally, set ∼ ∼ 3 H = CSpin(7)(ξ) = Spin(4) ×C2 Spin(3) = SU(2) /{±(I,I,I)}

(the central product). Then in the same way, the action of NH (W )/W on ∗ 2 H (BW ; F2) can be identiﬁed with that of GL2′ (F2).

Proposition 4.7 The inclusions BW −−−−−→ BH −−−−−→ BSpin(7) −−−−−→ BDI(4) ∗ as deﬁned above, together with the identiﬁcation H (BW ; F2)= F2[x, y, z, w], induce isomorphisms

def ∗ GL4(F2) A = H (BDI(4); F2)= F2[x, y, z, w] = F2[a8, a12, a14, a15] 3 def ∗ GL (F2) B = H (BSpin(7); F2)= F2[x, y, z, w] 1 = F2[b4, b6, b7, b8] (∗) def GL2 (F ) C ∗ 2′ 2 ′ ′′ = H (BH; F2)= F2[x, y, z, w] = F2[c2, c3, c4, c4] ;

Geometry & Topology, Volume 6 (2002) 962 Ran Levi and Bob Oliver where

a8 = D1(x, y, z, w), a12 = D2(x, y, z, w),

a14 = D3(x, y, z, w), a15 = D4(x, y, z, w); b4 = D1(x,y,z), b6 = D2(x,y,z), b7 = D3(x,y,z), b8 = (w + α); α∈hYx,y,zi and ′ ′′ c2 = D1(x,y), c3 = D2(x,y), c4 = (z + α), c4 = (w + α). α∈hYx,yi α∈hYx,yi Furthermore,

3 (a) the natural action of Σ3 on H =∼ SU(2) /{±(I,I,I)} induces the action C ′ ′′ ′ ′′ on which ﬁxes c2, c3 and permutes {c4, c4 , c4 + c4}; and (b) the above variables satisfy the relations 2 2 2 a8 = b8 + b4 a12 = b8b4 + b6 a14 = b8b6 + b7 a15 = b8b7 ′ 2 ′ 2 ′ ′′ ′ ′′ b4 = c4 + c2 b6 = c2c4 + c3 b7 = c3c4 b8 = c4(c4 + c4) .

∗ Proof The formulas for A = H (BDI(4); F2) are shown in [9]. From [9, Lemmas 3.10 and 3.11], we see there are (some) identiﬁcations

3 F 2 F ∗ GL1( 2) ∗ GL2′ ( 2) H (BSpin(7); F2) =∼ F2[x,y,z,w] and H (BH; F2) =∼ F2[x,y,z,w] .

From the explicit choices of subgroups W ≤ H ≤ Spin(7) as described above (and by the descriptions in Proposition A.8 of the automorphism groups), the ∗ ∗ images of H (BSpin(7); F2) and H (BH; F2) in F2[x, y, z, w] are seen to be contained in the rings of invariants, and hence these isomorphisms actually are equalities as claimed. We next prove the equalities in (∗) between the given rings of invariants and polynomial algebras. The following argument was shown to us by Larry Smith. If k is a field and V is an n–dimensional vector space over k, then a system of parameters in the polynomial algebra k[V ] is a set of n homogeneous elements f1,...,fn such that k[V ]/(f1,...,fn) is finite dimensional over k. By [21, Proposition 5.5.5], if V is an n–dimensional k[G]–representation, and G f1,...,fn ∈ k[V ] is a system of parameters the product of whose degrees G is equal to |G|, then k[V ] is a polynomial algebra with f1,...,fn as generators. By [21, Proposition 8.1.7], F2[x, y, z, w] is a free finitely generated

Geometry & Topology, Volume 6 (2002) Construction of 2–local finite groups 963 module over the ring generated by its Dickson invariants (this holds for polynomial algebras over any Fp ), and thus F2[x, y, z, w]/(a8, a12, a14, a15) is finite. (This can also be shown directly using the relation in Lemma 4.6.) So assuming the relations in point (b), the quotients F2[x, y, z, w]/(b4, b6, b7, b8) and ′ ′′ F2[x, y, z, w]/(c2, c3, c4, c4) are also finite. In each case, the product of the degrees of the generators is clearly equal to the order of the group in question, and this finishes the proof of the last equality in the second and third lines of (∗).

It remains to prove points (a) and (b). Using Lemma 4.6, the ci are expressed as polynomials in x, y, z, w as follows:

2 2 c2 = D1(x, y)= x + xy + y

c3 = D2(x, y)= xy(x + y) ′ 2 4 2 4 2 (1) c4 = D1(x,y,z)+ D1(x, y) = z + z D1(x, y)+ zD2(x, y)= z + z c2 + zc3 ′′ 2 4 2 4 2 c4 = D1(x, y, w)+ D1(x, y) = w + w D1(x, y)+ wD2(x, y)= w + w c2 + wc3 .

In particular,

′ ′′ 4 2 c4 +c4 = (z +w) +(z +w) D1(x,y)+(z +w)D2(x,y)= (z +w +α). (2) α∈hYx,yi Furthermore, by (1), we get

1 Sq (c2)= c3 1 1 ′ 1 ′′ Sq (c3)= Sq (c4)= Sq (c4) = 0 2 2 2 3 Sq (c3)= x y (x + y)+ xy(x + y) = c2c3 2 ′ 4 2 2 ′ Sq (c4)= z c2 + z c2 + zc2c3 = c2c4 (3) 3 ′ 1 ′ ′ Sq (c4)= Sq (c2c4)= c3c4 2 ′′ ′′ Sq (c4)= c2c4 3 ′′ ′′ Sq (c4)= c3c4.

3 The permutation action of Σ3 on H =∼ SU(2) /{±(I,I,I)} permutes the three elements ζ,ξ,ζ + ξ of Z(H) ⊆ W , and thus (via the identification V = W ∗ described above) induces the identity on x,y ∈ V and permutes the elements {z,w,z + w} modulo hx,yi. Hence the induced action of Σ3 on 2 GL ′ (F2) C = F2[V ] 2 is the restriction of the action on F2[V ]= F2[x, y, z, w] which fixes x,y and permutes {z,w,z +w}. So by (1) and (2), we see that this action ′ ′′ ′ ′′ fixes c2, c3 and permutes the set {c4, c4, c4 + c4}. This proves (a).

Geometry & Topology, Volume 6 (2002) 964 Ran Levi and Bob Oliver

It remains to prove the formulas in (b). From (1) and (3) we get ′ 2 b4 = D1(x,y,z)= c4 + c2, 2 ′ 2 b6 = D2(x,y,z)= Sq (b4)= c2c4 + c3, 1 ′ b7 = D3(x,y,z)= Sq (b6)= c3c4. Also, by (1) and (2),

′′ ′ ′′ b8 = (w + α)= (w + α) · (w + z + α) = c4(c4 + c4). α∈hYx,y,zi α∈hYx,yi α∈hYx,yi This proves the formulas for the bi in terms of ci . Finally, we have 2 a8 = D1(x, y, z, w)= b8 + b4, 4 2 4 ′′ ′ ′′ ′ 2 2 a12 = D2(x, y, z, w)= Sq (b8 + b4)= Sq (c4(c4 + c4) + (c4 + c2) ) ′ ′′ ′ ′′ 2 ′′ ′ ′′ 2 ′ 2 4 2 = c4c4(c4 + c4)+ c2c4(c4 + c4)+ c2c4 + c3 = b8b4 + b6 2 ′ ′′ ′ ′′ 2 ′′ ′ ′′ 2 ′ 2 a14 = D3(x, y, z, w)= Sq (a12)= c2c4c4(c4 + c4)+ c3c4(c4 + c4)+ c3c4 2 = b8b6 + b7 1 ′ ′′ ′ ′′ a15 = D4(x, y, z, w)= Sq (a14)= c3c4c4(c4 + c4)= b8b7 ; and this ﬁnishes the proof of the proposition.

C ′ Lemma 4.8 Let κ ∈ Aut( ) be the algebra involution which exchanges c4 and ′′ C c4 and leaves c2 and c3 ﬁxed. An element of will be called “κ–invariant” if it is ﬁxed by this involution. Then the following hold:

(a) If β ∈ B is κ–invariant, then β ∈ A. B ′ i ′ i ′ A (b) If β ∈ is such that β·c4 is κ–invariant, then β = β ·b8 for some β ∈ .

Proof Point (a) follows from Proposition 4.7 upon regarding A, B, and C as 3 2 the ﬁxed subrings of the groups GL4(F2), GL1(F2) and GL2′ (F2) acting on F2[x, y, z, w], but also follows from the following direct argument. Let m be the degree of β as a polynomial in b8 ; we argue by induction on m. Write m β = β0 + b8 ·β1 , where β1 ∈ F2[b4, b6, b7], and where β0 has degree < m (as ′ a polynomial in b8 ). If m = 0, then β = β1 ∈ F2[b4, b6, b7] ⊆ F2[c2, c3, c4], and hence β ∈ F2[c2, c3] since it is κ–invariant. But from the formulas in Proposition 4.7(b), we see that F2[b4, b6, b7] ∩ F2[c2, c3] contains only constant polynomials (hence it is contained in A). ′ ′′ Now assume m ≥ 1. Then, expressed as a polynomial in c2, c3, c4, c4 , the ′′ ′′2m largest power of c4 which occurs in β is c4 . Since β is κ–invariant, the

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 965

′ ′ 2m highest power of c4 which occurs is c4 ; and hence by Proposition 4.7(b), the total degree of each term in β1 (its degree as a polynomial in b4, b6, b7 ) is at r s t most m. So for each term b4b6b7 in β1 , r s t m m−r−s−t r s t b4b6b7b8 − a8 a12a14a15 is a sum of terms which have degree < m in b8 , and thus lies in A by the induction hypothesis.

′ i ′ i To prove (b), note ﬁrst that since β·c4 is κ–invariant and divisible by c4 , it ′′i ′′i must also be divisible by c4 , and hence c4 |β. Furthermore, by Proposition B ′ 4.7, all elements of as well as c4 are invariant under the involution which ′ ′′ ′ ′′ ′ ′′ i ′′ ′ ′′ ﬁxes c4 and sends c4 7→ c4 + c4 . Thus (c4 + c4) |β. Since b8 = c4(c4 + c4), we ′ i B can now write β = β ·b8 for some β ∈ . Finally, since ′ i ′ ′ i ′′i ′ ′′ i β·c4 = β ·c4 ·c4 ·(c4 + c4) is κ–invariant, β′ is also κ–invariant, and hence β′ ∈ A by (a).

2 GL ′ (F2) Note that C3 ≤ Σ3 = GL2(F2) act on C = F2[x, y, z, w] 2 : via the action 2 2 ′ ′′ of the group GL2(F2)/GL2′ (F2), or equivalently by permuting c4 , c4 , and ′ ′′ A B CC3 c4 + c4 (and ﬁxing c2, c3 ). Thus = ∩ , since GL4(F2) is generated 3 2 by the subgroups GL1(F2) and GL2(F2). This is also shown directly in the following lemma.

Proposition 4.9 The following square is a pullback square, where all maps are induced by inclusions between the subrings of F2[x, y, z, w]:

Ω∗ → Ω∗ C3 A/F2 C/F2 ↓ ↓ ∗ ∗ ΩB/F2 → ΩC/F2 .

Proof Let κ be the involution of Lemma 4.8: the algebra involution of C ′ ′′ which exchanges c4 and c4 and leaves c2 and c3 fixed. By construction, all elements in the image of Ω∗ are invariant under the involution which fixes B/F2 c′ (and c , c ), and sends c′′ to c′ + c′′ . Hence elements in the image of Ω∗ 4 2 3 4 4 4 B/F2 are fixed by C3 if and only if they are fixed by Σ3 , if and only if they are κ– invariant. So it will suffice to show that all of the above maps are injective, and that all κ–invariant elements in the image of Ω∗ lie in the image of Ω∗ . B/F2 A/F2 The injectivity is clear, and the square is a pullback for Ω0 by Lemma 4.8. −/F2

Geometry & Topology, Volume 6 (2002) 966 Ran Levi and Bob Oliver

Fix a κ–invariant element

ω = P1 db4 + P2 db6 + P3 db7 + P4 db8 ′ ′ ′ ′′ ′′ ′ 1 (1) = P2c4 dc2 + P3c4 dc3 + P4c4 dc4 + (P1 + P2c2 + P3c3 + P4c4) dc4 ∈ ΩB/F2 , where Pi ∈ B for each i. By applying κ to (1) and comparing the coefficients of ′ ′ dc2 and dc3 , we see that P2c4 and P3c4 are κ–invariant. Also, upon comparing ′ the coefficients of dc4 , we get the equation ′′ ′′ P1 + P2c2 + P3c3 + P4c4 = κ(P4)c4. (2) ′ ′ ′ ′ A So by Lemma 4.8, P2 = P2b8 and P3 = P3b8 for some P2, P3 ∈ . Upon subtracting ′ ′ ′ ′ P2 da14 + P3 da15 = P2 db6 + P3 db7 + (P2b6 + P3b7) db8 from ω and introducing an appropriate modification to P4 , we can now assume that P2 = P3 = 0. With this assumption and (2), we have ′′ ′ ′′ P1 + P4c4 = κ(P4c4)= κ(P4)·c4, so that ′ ′ ′′ P1c4 = (P4 + κ(P4))c4c4 (3) ′ ′ A is κ–invariant. This now shows that P1 = P1b8 for some P1 ∈ , and upon ′ subtracting P1 da12 from ω we can assume that P1 = 0. This leaves ω = P4 db8 = P4 da8 . By (3) again, P4 is κ–invariant, so P4 ∈ A by Lemma 4.8 again, and thus ω ∈ Ω1 . A/F2 The remaining cases are proved using the same techniques, and so we sketch them more briefly. To prove the result in degree two, fix a κ–invariant element

ω = P1 db4 db6 + P2 db4 db7 + P3 db4 db8 + P4 db6 db7 + P5 db6 db8 + P6 db7 db8 ′ 2 ′ ′ ′ ′′ ′ ′ 2 ′′ = P4c4 dc2 dc3 + (P1c4 + P4c3c4 + P5c4c4) dc2 dc4 + P5c4 dc2 dc4 ′ ′ ′ ′′ ′ ′ 2 ′′ + (P2c4 + P4c2c4 + P6c4c4) dc3 dc4 + P6c4 dc3 dc4 ′ ′ ′ ′ ′′ 2 + (P3c4 + P5c2c4 + P6c3c4) dc4 dc4 ∈ ΩB/F2 . ′ 2 Using Lemma 4.8, we see that P4 = P4b8 , and hence can assume that P4 = 0. One then eliminates P1 and P2 , then P5 and P6 , and ﬁnally P3 . If

ω = P1 db4 db6 db7 + P2 db4 db6 db8 + P3 db4 db7 db8 + P4 db6 db7 db8 ′ 2 ′ 2 ′′ ′ ′ 2 ′ 2 ′ ′′ = (P1c4 + P4c4 c4) dc2 dc3 dc4 + (P2c4 + P4c3c4 ) dc2 dc4 dc4 + (P c′ 2 + P c c′ 2) dc dc′ dc′′ + P c′ 3 dc dc dc′′ ∈ Ω3 3 4 4 2 4 3 4 4 4 4 2 3 4 B/F2

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 967

is κ–invariant, then we eliminate successively P1 , then P4 , then P2 and P3 . Finally, if ω = P db db db db = P c′ 3 dc dc dc′ dc′′ ∈ Ω4 4 6 7 8 4 2 3 4 4 B/F2 ′ 3 ′ A is κ–invariant, then P = P b8 for some P ∈ by Lemma 4.8 again, and so ω = P ′ da da da da ∈ Ω4 . 8 12 14 15 A/F2

A Appendix : Spinor groups over ﬁnite ﬁelds

Let F be any ﬁeld of characteristic 6= 2. Let V be a vector space over F , and let b: V −−→ F be a nonsingular quadratic form. As usual, O(V, b) denotes the group of isometries of (V, b), and SO(V, b) the subgroup of isometries of determinant 1. We will be particularly interested in elementary abelian 2– subgroups of such orthogonal groups.

Lemma A.1 Fix an elementary abelian 2–subgroup E ≤ O(V, b). For each irreducible character χ ∈ Hom(E, {±1}), let Vχ ⊆ V denote the corresponding eigenspace: the subspace of elements v ∈ V such that g(v) = χ(g)·v for all g ∈ E . Then the restriction of b to each subspace Vχ is nonsingular, and V is the orthogonal direct sum of the Vχ .

Proof Elementary.

We give a very brief sketch of the definition of spinor groups via Clifford algebras; for more details we refer to [8, section II.7] or [2, section 22]. Let T (V ) denote the tensor algebra of V , and set C(V, b)= T (V )/h(v ⊗ v) − b(v)i : the Clifford algebra of (V, b). To simplify the notation, we regard F as a subring of C(V, b), and V as a subgroup of its additive group; thus the class of v1 ⊗···⊗ vk will be written v1· · ·vk . Note that vw + wv =0 if v, w ∈ V and v ⊥ w. Hence if dimF (V ) = n, and {v1, . . . , vn} is an orthogonal basis, then the set of 1 and all vi1 · · · vik for i1 < · · · < ik (1 ≤ k ≤ n) is an F –basis for C(V, b).

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Write C(V, b) = C0 ⊕ C1 , where C0 and C1 consist of classes of elements of even or odd degree, respectively. Let G ≤ C(V, b)∗ denote the group of invertible elements u such that uV u−1 = V , and let π : G −−→ O(V, b) be the homomorphism −1 (v 7→ −uvu ) if u ∈ C1 π(u)= −1 ((v 7→ uvu ) if u ∈ C0 .

In particular, for any nonisotropic element v ∈ V (ie, b(v) 6= 0), v ∈ G and π(v) is the reﬂection in the hyperplane v⊥ . By [8, section II.7], π is surjective and Ker(π)= F ∗ .

Let J be the antiautomorphism of C(V, b) induced by the antiautomorphism v1 ⊗···⊗ vk 7→ vk ⊗···⊗ v1 of T (V ). Since O(V, b) is generated by hyperplane reﬂections, G is generated by F ∗ and nonisotropic elements v ∈ V . In particular, for any u = λ·v1 · · · vk ∈ G,

2 2 ∗ J(u)·u = λ · vk · · · v1 · v1 · · · vk = λ · b(v1) · · · b(vk) ∈ F = Ker(π); implying that π(J(u)) = π(u)−1 for all u ∈ G. There is thus a homomorphism

θ : G −−−−−−−→ F ∗ deﬁned by θ(u)= u·J(u).

In particular,eθ(λ) = λ2 for λ ∈ F ∗ ≤ G, whilee for any set of nonisotropic elements v1, . . . , vk of V , e θ(v1 · · · vk) = (v1 · · · vk)(vk · · · v1)= b(v1) · · · b(vk).

Hence θ factorse through a homomorphism

∗ ∗2 ∗ 2 ∗ e θV,b : O(V, b) −−−−−−−→ F /F = F /{u | u ∈ F }, called the spinor norm.

+ −1 Set G = π (SO(V, b)) = G ∩ C0 , and deﬁne

Spin(V, b) = Ker(θ|G+ ) and Ω(V, b) = Ker(θV,b|SO(V,b)).

In particular, Ω(V, b) hase index 2 in SO(V, b) if F is a ﬁnite ﬁeld, and Ω(V, b)= SO(V, b) if F is algebraically closed (all units are squares). We thus get a commutative diagram

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1 1 1 / / λ7→λ2 / / 1 / {±1} / F ∗ / F ∗2 / 1

/ / θ / / (A.2) 1 / Spin(V, b) / G+ / F ∗ / 1 e π / / θV,b / / 1 / Ω(V, b) / SO(V, b) / F ∗/F ∗2 / 1 1 1 1 where all rows and columns are short exact, and where all columns are central extensions of groups. If dim(V ) ≥ 3 (or if dim(V ) = 2 and the form b is hyperbolic), then Ω(V, b) is the commutator subgroup of SO(V, b) [8, section II.8]. The following lemma follows immediately from this description of Spin(V, b), together with the analogous description of the corresponding spinor group over the algebraic closure of F .

Lemma A.3 Let F be the algebraic closure of F , and set V = F ⊗F V and b b b b = IdF ⊗ . Then Spin(V, ) is the subgroup of Spin(V , ) consisting of those elements ﬁxed by all Galois automorphisms ψ ∈ Gal(F /F ).

For any nonsingular quadratic form b on a vector space V , the discriminant of b (or of V ) is the determinant of the corresponding symmetric bilinear form B, related to b by the formulas b 1 b b b (v)= B(v, v) and B(v, w)= 2 (v + w) − (v) − (w) . Note that the discriminant is well deﬁned only modulo squares in F ∗. When W ⊆ V is a subspace, then we deﬁne the discriminant of W to mean the discriminant of b|W . In what follows, we say that the discriminant of a quadratic form is a square or a nonsquare to mean that it is the identity or not in the quotient group F ∗/F ∗2 .

Lemma A.4 Fix an involution x ∈ SO(V, b), and let V = V+ ⊕ V− be its eigenspace decomposition. Then the following hold.

(a) x ∈ Ω(V, b) if and only if the discriminant of V− is a square.

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(b) If x ∈ Ω(V, b), then it lifts to an element of order 2 in Spin(V, b) if and only if dim(V−) ∈ 4Z. (c) If x ∈ Ω(V, b), and if α ∈ Ω(V, b) is such that [x, α] = 1, then α = α+ ⊕ α− , where α± ∈ O(V±, b). Also, the liftings of x and α commute in Spin(V, b) if and only if det(α−) = 1.

Proof Let {v1, . . . , vk} be an orthogonal basis for V− (k is even). Then x = π(v1· · ·vk) in the above notation, since π(vi) is the reﬂection in the hyperplane ⊥ vi . Hence by the commutativity of Diagram (A.2), b b b ∗2 θV,b(x) ≡ (v1)· · · (vk) = det( |V− ) (mod F ).

Thus x ∈ Ω(V, b) = Ker(θV,b) if and only if V− has square discriminant.

In particular, if x ∈ Ω(V, b), then the product of the b(vi) is a square, and hence (upon replacing v1 by a scalar multiple) we can assume that b(v1)· · ·b(vk)=1. def Then x = v1 · · · vk ∈ Spin(V, b) = Ker(θ). Since vw = −wv in the Cliﬀord 2 algebra whenever v ⊥ w, and since (vi) = b(vi) for each i, e e 2 k(k−1)/2 2 2 k(k−1)/2 1 if k ≡ 0 (mod 4) x = (−1) ·(v1) · · ·(vk) = (−1) = (−1 if k ≡ 2 (mod 4) . Thise proves (b).

It remains to prove (c). The ﬁrst statement (α = α+ ⊕α− ) is clear. Fix liftings ∗ α± ∈ C(V±, b) . Rather than deﬁning the direct sum of an element of C(V+, b) ∗ with an element of C(V−, b), we regard the groups C(V±, b) as (commuting) ∗ subgroupse of C(V, b) , and set

α = α+ ◦ α− = α− ◦ α+ ∈ Spin(V, b).

Let x = v1· · ·vk be as above. Clearly, x commutes with all elements of C(V+, b). Since e e e e e k−1 e (v1· · ·vk)·vi = (−1) ·vei·(v1· · ·vk)= −vi·(v1· · ·vk) i for i = 1,...,k, we have x·β = (−1) ·β·x for all β ∈ Ci(V−, b) (i = 0, 1). In particular, since [α+, α−]=1, [x, α] = [x, α−] = det(α−), and this ﬁnishes the proof. e e e e e e e e We will need explicit isomorphisms which describe Spin3(F ) and Spin4(F ) in terms of SL2(F ). These are constructed in the following proposition, where 0 M2 (F ) denotes the vector space of matrices of trace zero. Note that the de- 0 terminant is a nonsingular quadratic form on M2(F ) and on M2 (F ), in both cases with square discriminant.

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Proposition A.5 Deﬁne 0 ρ3 : SL2(F ) −−−−−−→ Ω(M2 (F ), det) and ρ4 : SL2(F ) × SL2(F ) −−−−−−→ Ω(M2(F ), det) by setting −1 −1 ρ3(A)(X)= AXA and ρ4(A, B)(X)= AXB .

Then ρ3 and ρ4 are both epimorphisms, and lift to unique isomorphisms

ρ3 0 SL2(F ) −−−−−−→ Spin(M2 (F ), det) =∼ e and ρ4 SL2(F ) × SL2(F ) −−−−−−→ Spin(M2(F ), det). =∼ e Proof See [24, pages 142, 199] for other ways of defining these isomorphisms. By Lemma A.3, it suffices to prove this (except for the uniqueness of the lifting) when F is algebraically closed. In particular, 0 0 Ω(M2 (F ), det) = SO(M2 (F ), det) and Ω(M2(F ), det) = SO(M2(F ), det) in this case. For general V and b, the group SO(V, b) is generated by reflections fixing nonisotropic subspaces (ie, of nonvanishing discriminant) of codimension 2 (cf [8, section II.6(1)]). Hence to see that ρ3 and ρ4 are surjective, it suffices to show that such elements lie in their images. A codimension 2 reflection in 0 SO(M2 (F ), det) is of the form RX (the reflection fixing the line generated by 0 X ) for some X ∈ M2 (F ) which is nonisotropic (ie, det(X) 6= 0). Since F 2 is algebraically closed, we can assume X ∈ SL2(F ). Then X = −I (since Tr(X) = 0 and det(X) = 1), and RX = ρ3(X) since it has order 2 and fixes X . Thus ρ3 is onto. Similarly, any 2–dimensional nonisotropic subspace W ⊆ V has an orthonormal basis {Y,Z}, and ZY −1 and Y −1Z have trace zero (since they are orthogonal to the identity matrix) and determinant one. Hence their square is −I , and −1 −1 one repeats the above argument to show that RW = ρ4(ZY ,Y Z). So ρ4 is onto.

The liftings ρm exist and are unique since SL2(F ) is the universal central extension of PSL2(F ) (or universal among central extensions by 2–groups if F = F3 ). e

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We now restrict to the case F = Fq where q is an odd prime power. We refer to [2, section 21] for a description of quadratic forms in this situation, and the notation for the associated orthogonal groups. If n is odd and b is any nonsingular n quadratic form on Fq , then every nonsingular quadratic form is isomorphic to b ∗ n b n b u for some u ∈ Fq , and hence one can write SOn(q)= SO(Fq , )= SO(Fq , u ) without ambiguity. If n is even, then there are exactly two isomorphism classes n + n b b of quadratic forms on Fq ; and one writes SOn (q) = SO(Fq , ) when is the hyperbolic form (equivalently, has discriminant (−1)n/2 modulo squares), and − n b b SOn (q) = SO(Fq , ) when is not hyperbolic (equivalently, has discriminant n/2 ∗ (−1) ·u for u ∈ Fq not a square). This notation extends in the obvious way ± ± to Ωn (q) and Spinn (q). The following lemma does, in fact, hold for for orthogonal representations over arbitrary fields of characteristic 6= 2. But to simplify the proof (and since we were unable to find a reference), we state it only in the case of finite fields.

Lemma A.6 Assume F = Fq , where q is a power of an odd prime. Let V be an F –vector space, and let b be a nonsingular quadratic form on V . Let P ≤ O(V, b) be a 2–subgroup which is orthogonally irreducible; ie, such that V has no splitting as an orthogonal direct sum of nonzero P –invariant subspaces. Then dimF (V ) is a power of 2; and if dim(V ) > 1 then b has square discriminant.

n Proof This means showing that each orthogonal group O(Fq , b), such that either n is not a power of 2, or n = 2k ≥ 2 and the quadratic form b has ± ± nonsquare discriminant, contains some subgroup Om(q) × On−m(q) (for 0 < m

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Assume n is not a power of 2, and write n = 2k + m where 0

ǫ m−1 2(2k+i) 2k+m 2k |O (q)| k+1 q − 1 q − ǫ q + 1 2n = qm2 · · · , + ǫ q2i − 1 qm − ǫ 2 |O2k+1 (q)|·|O2m(q)| ! ! ! Yi=1 k and each factor is invertible in Z(2) . When n = 2m = 2 and k ≥ 1, then − O2n(q) is the orthogonal group for the quadratic form with nonsquare discriminant, and

− m−1 2(m+i) 2m |O2n(q)| 2m2 q − 1 q + 1 + − = q · 2i · , |O2m(q)|·|O2m(q)| q − 1 ! 2 Yi=1 and again each factor is invertible in Z(2) . Finally, |Oǫ(q)| q − ǫ 2 = |O1(q)|·|O1(q)| 2 is odd whenever q ≡ 1 (mod 4) and ǫ = −1, or q ≡ 3 (mod 4) and ǫ = +1; and 2 these are precisely the cases where the quadratic form on Fq has nonsquare discriminant.

We must classify the conjugacy classes of those elementary abelian 2–subgroups of Spin7(q) which contain its center. The following deﬁnition will be useful when doing this.

7 b Deﬁnition A.7 Fix an odd prime power q. Identify SO7(q)= SO(Fq, ) and 7 b b Spin7(q) = Spin(Fq, ), where is a nonsingular quadratic form with square discriminant. An elementary abelian 2–subgroup of SO7(q) or of Spin7(q) will be called of type I if its eigenspaces all have square discriminant (with respect to b), and of type II otherwise. Let En be the set of elementary abelian 2– ∼ subgroups in Spin7(q) which contain Z(Spin7(q)) = C2 and have rank n. Let I II En and En be the subsets of En consisting of those subgroups of types I and II, respectively.

In the following two propositions, we collect together the information which will be needed about elementary abelian 2–subgroups of Spin7(q). We ﬁx b ∼ 7 b Spin7(q) = Spin(V, ), where V = Fq , and is a nonsingular quadratic form with square discriminant. Let z ∈ Z(Spin7(q)) be the generator. For any subgroup H ≤ Spin7(q) or any element g ∈ Spin7(q), let H and g denote their images in Ω7(q) ≤ SO7(q). For each elementary abelian 2–subgroup E ≤ Spin7(q), and each character χ ∈ Hom(E, {±1}), Vχ ⊆ V denotes the

Geometry & Topology, Volume 6 (2002) 974 Ran Levi and Bob Oliver

eigenspace of χ (and V1 denotes the eigenspace of the trivial character). Also (when z ∈ E ), Aut(E, z) denotes the group of all automorphisms of E which send z to itself.

Proposition A.8 For any odd prime power q, the following table describes I II the numbers of Spin7(q)–conjugacy classes in each of the sets En and En , the dimensions and discriminants of the eigenspaces of subgroups in these sets, and indicates in which cases Aut (E) contains all automorphisms which ﬁx Spin7(q) z. I I II I II Set of subgroups E2 E3 E3 E4 E4 Nr. conj. classes 1 1 1 2 1

dim(V1) 3 1 0

dim(Vχ), χ 6= 1 4 2 1

discr(V1, b) square square nonsq. — —

discr(Vχ, b), χ 6= 1 square square nonsq. square both

AutSpin7(q)(E) = Aut(E, z) yes yes yes yes no

There are no subgroups in E2 of type II, and no subgroups of rank ≥ 5. Fur- thermore, we have:

(a) For all E ∈E , C (E)= E . 4 Spin7(q) ′ I ′ −1 ′ (b) If E, E ∈E4 , then E = gEg for some g ∈ SO7(q), and E and E are Spin7(q)–conjugate if and only if g ∈ Ω7(q). II (c) If E ∈ E4 , then there is a unique element 1 6= x = x(E) ∈ E with the property that for 1 6= χ ∈ Hom(E, {±1}), Vχ has square discriminant if χ(x) = 1 and nonsquare discriminant if χ(x) = −1. Also, the image of

AutSpin7(q)(E) in Aut(E) is the group of all automorphisms which send x to itself; and if X ≤ E denotes the inverse image of hxi ≤ E , then Aut (E) contains all automorphisms of E which are the identity on Spin7(q) X and the identity modulo hzi.

(d) If E ∈E3 , then CSpin7(q)(E)= A⋊C2 , where A is abelian and C2 acts on A by inversion. If E ∈ EII , then the Sylow 2–subgroups of C (E) 3 Spin7(q) are elementary abelian of rank 4 (and type II).

Proof Write Spin = Spin7(q) for short. Fix an elementary abelian subgroup E ≤ Spin such that z ∈ E .

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Step 1 We ﬁrst show that rk(E) ≤ 4, and that the dimensions of the eigenspaces Vχ for χ ∈ Hom(E, {±1}) are as described in the table. By Lemma A.4, every involution in E has a 4–dimensional (−1)–eigenspace. In particular, if rk(E) = 2, (E =∼ C2 ), then dim(Vχ) = 4 for 1 6= χ ∈ Hom(E, {±1}), while dim(V1)=3. Now assume rk(E) = n for some n > 2. Assume we have shown, for all ′ ′ E ∈En−1 , that the eigenspace of the trivial character of E is r–dimensional. For each 1 6= χ ∈ Hom(E, {±1}), let Eχ ∈ En−1 be the subgroup such that Eχ = Ker(χ); then V1 ⊕ Vχ is the eigenspace of the trivial character of Eχ = Ker(χ), and thus dim(V1) + dim(Vχ)= r. Hence all nontrivial characters of E have eigenspaces of the same dimension. Since there are 2n−1 − 1 nontrivial n−1 characters of E , we have dim(V1) + (2 − 1) dim(Vχ) = 7, and these two equations completely determine dim(V1) and dim(Vχ). Using this procedure, the dimensions of the eigenspaces are shown inductively to be equal to those given by the table. Also, this shows that rk(E) ≤ 4, since otherwise rk(E) ≥ 4, so the Vχ for χ 6= 1 must be trivial (they cannot all have dimension ≥ 1), so E acts on V via the identity, which contradicts the assumption that E ≤ Spin7(q). II Step 2 We next show that E2 = ∅, describe the discriminants of the eigenspaces of characters of E for E ∈ En (for all n), and show that subgroups of rank 4 are self centralizing. In particular, this proves (a) together with the ﬁrst statement of (c).

If E ∈E2 , then E = hz, gi for some noncentral involution g ∈ Spin7(q), and the eigenspaces of E = hgi have square discriminant by Lemma A.4(a) (and since II the ambient space V has square discriminant by assumption). Thus E2 = ∅.

If E ∈ E3 , then the sum of any two eigenspaces of E is an eigenspace of g for some g ∈ Erhzi. Hence the sum of any two eigenspaces of E has square I discriminant, so either all of the eigenspaces have square discriminant (E ∈E3 ), II or all of the eigenspaces have nonsquare discriminant (E ∈E3 ). Assume rk(E) = 4. We have seen that all eigenspaces of E are 1–dimensional.

By Lemma A.4(c), for each a ∈ CSpin7(q)(E), a(Vχ)= Vχ for each χ 6= 1, and since dim(Vχ) = 1 it must act on each Vχ via ± Id. Thus a ∈ Ω7(q) has order 2; let V± be its eigenspaces. Then dim(V−) is even since det(a) = 1, and V− has square discriminant by Lemma A.4(a). If dim(V−) = 4, then |a| = 2 (Lemma A.4(b)), and hence a ∈ E since otherwise hE, ai would have rank 5. If dim(V−) = 2, then V− is the sum of the eigenspaces of two distinct characters

χ1,χ2 of E , there is some g ∈ E such that χ1(g) 6= χ2(g), hence det(g|V− ) =

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χ1(g)χ2(g) = −1, so [g, a] = z by Lemma A.4(c), and this contradicts the assumption that [a, E] = 1. If dim(V−) = 6, then V− is the sum of the eigenspaces of all but one of the nontrivial characters of E , and this gives a similar contradiction to the assumption [a, E] = 1. Thus, C (E)= E . Spin7(q) II Now assume that E ∈ E4 , and let x ∈ O7(q) be the element which acts via − Id on eigenspaces with nonsquare discriminant, and via the identity on those with square discriminant. Since b has square discriminant on V , the number of eigenspaces of E on which the discriminant is nonsquare is even, so x ∈ Ω7(q) by Lemma A.4(a), and lifts to an element x ∈ Spin7(q). Also, for each g ∈ E , the (−1)–eigenspace of g has square discriminant (Lemma A.4(a) again), hence contains an even number of eigenspaces of E of nonsquare discriminant, and by Lemma A.4(c) this shows that [g,x] = 1. Thus x ∈ CSpin7(q)(E) = E , and this proves the ﬁrst statement in (c).

Step 3 We next check the numbers of conjugacy classes of subgroups in each I II of the sets En and En , and describe AutSpin(E) in each case. This ﬁnishes the proof of (b) and (c), and of all points in the above table. From the above description, we see immediately that if E and E′ have the same rank and type, then any isomorphism α ∈ Iso(E, E′), such that α(x(E)) = ′ ′ II ′ x(E ) if E, E ∈E4 , has the property that for all χ ∈ Hom(E , {±1}), Vχ and Vχ◦α have the same dimension and the same discriminant (modulo squares). Hence for any such α, there is an element g ∈ O7(q) such that g(Vχ◦α) = Vχ ′ for all χ; and α = cg ∈ Iso(E, E ) for such g. Upon replacing g by −g if necessary, we can assume that g ∈ SO7(q). This shows that

′ ′ E, E have the same rank and type =⇒ E and E are SO7(q)–conjugate (1) and also that II Aut(E) if E∈E / 4 AutSO7(q)(E)= II (2) (Aut(E, x(E)) if E ∈E4 .

We next claim that I E∈E / 4 =⇒ ∃γ ∈ SO7(q)rΩ7(q) such that [γ, E] = 1 . (3) ′ To prove this, choose 1–dimensional nonisotropic summands W ⊆ Vχ and W ⊆ Vψ , where χ, ψ are two distinct characters of E , and where W has square ′ discriminant and W has nonsquare discriminant. Let γ ∈ SO7(q) be the involution with (−1)–eigenspace W ⊕ W ′ . Then [γ, E] = 1, since γ sends

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each eigenspace of E to itself, and γ∈ / Ω7(q) since its (−1)–eigenspace has nonsquare discriminant (Lemma A.4(a)).

If E has rank 4 and type I, then Aut(E) =∼ GL3(F2) is simple, and in particular has no subgroup of index 2. Hence by (2), each element of Aut(E) is induced by conjugation by an element of Ω7(q). Also, if g ∈ SO7(q) centralizes E , then g(Vχ) = Vχ for all χ ∈ Hom(E, {±1}), g acts via − Id on an even number of eigenspaces (since it has determinant +1), and hence g ∈ Ω7(q) by Lemma A.4(a). Thus I E ∈E4 =⇒ NSO7(q)(E) ≤ Ω7(q) (4)

I If E∈ / E4 , then by (3), for any g ∈ SO7(q), there is γ ∈ SO7(q)rΩ7(q) such ′ that cg|E = cgγ|E , and either g or gγ lies in Ω7(q). Thus IsoSO7(q)(E, E ) = ′ ′ IsoΩ7(q)(E, E ) for any E . Together with (1), this shows that E is Spin– conjugate to all other subgroups of the same rank and type, and together with (2) it shows that

II Aut(E) if E∈E / 4 Im AutSpin(E) −−→ Aut(E) = II (5) (Aut(E, x(E)) if E ∈E4 . I If E ∈ E4 , then by (4) and (2), AutΩ7(q)(E) = AutSO7(q)(E) = Aut(E), and so (5) also holds in this case. Furthermore, for any g ∈ SO7(q)rΩ7(q), E and −1 gEg are representatives for two distinct Ω7(q)–conjugacy classes — since by (4), no element of the coset g·Ω7(q) normalizes E . We have now determined in all cases the number of conjugacy classes of subgroups of a given rank and type, and the image of AutSpin(E) in Aut(E). We I next claim that if rk(E) < 4 or E ∈E4 , then II E∈E / 4 =⇒ AutSpin(E) ≥ α ∈ Aut(E) α(z)= z, α ≡ Id (mod hzi) . (6) Together with (5), this will ﬁnish the proof that AutSpin(E) is the group of all automorphisms of E which send z to itself. We also claim that II E ∈E4 =⇒ AutSpin(E) ≥ α ∈ Aut(E) α|X = IdX , α ≡ Id (mod hzi) , (7) where X ≤ E denotes the inverse image of hx(E)i≤ E , and this will ﬁnish the proof of (c). We prove (6) and (7) together. Fix α ∈ Aut(E) (α 6= Id) which sends z to itself, II induces the identity on E , and such that α|X = IdX if E ∈ E4 . Then there is 1 6= χ ∈ Hom(E, {±1}) such that α(g) = g when χ(g) =1 and α(g) = zg

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when χ(g)= −1. Choose any character ψ such that Vψ 6=0 and Vψχ 6= 0, and ′ let W ⊆ Vψ and W ⊆ Vψχ be 1–dimensional nonisotropic subspaces with the II same discriminant (this is possible when E ∈ E4 since x(E) ∈ Ker(χ)). Let ′ g ∈ O7(q) be the involution whose (−1)–eigenspace is W ⊕W . Then g ∈ Ω7(q) by Lemma A.4(a), so g lifts to g ∈ Spin7(q), and using Lemma A.4(c) one sees that cg = α.

Step 4 It remains to prove (d). Assume E ∈E3 . Let 1= χ1,χ2,χ3,χ4 be the four characters of E , and set Vi = Vχi . Then dim(V1) = 1, dim(Vi) = 2 for i = 2, 3, 4, and the Vi either all have square discriminant or all have nonsquare 4 discriminant. For each g ∈ CSpin(E), we can write g = i=1 gi , where gi ∈ O(V , b ). For each pair of distinct indices i, j ∈ {2, 3, 4}, there is some g ∈ E i i L whose (−1)–eigenspace is Vi ⊕Vj , and hence det(gi ⊕gj) = 1 by Lemma A.4(c). This shows that the gi all have the same determinant. Let A ≤ CSpin(E) be the subgroup of index 2 consisting of those g such that det(gi) = 1 for all i. ∼ ∗ Now, SO1(Fq) = 1, while SO2(Fq) = Fq is the group of diagonal matrices −1 2 of the form diag(u, u ) with respect to a hyperbolic basis of Fq . Thus A ∗ 3 is contained in a central extension of C2 by (Fq) , and any such extension ∗ 3 ± is abelian since H2((Fq) ) = 0. Hence A is abelian. The groups O2 (q) are all dihedral (see [24, Theorem 11.4]). Hence for any g ∈ CSpin(E)rA, g has order 2 and (−1)–eigenspace of dimension 4 (its intersection with each Vi is 1–dimensional), and hence |g| = 2 by Lemma A.4(b). Thus all elements of CSpin(E)rA have order 2, so the centralizer must be a semidirect product of A with a group of order 2 which acts on it by inversion.

II Now assume that E ∈ E3 ; ie, that the Vi all have nonsquare discriminant. Then for i = 2, 3, 4, SO(Vi, bi) has order q ± 1, whichever is not a multiple of 4 (see [24, Theorem 11.4] again). Thus if g ∈ A ≤ CSpin(E) has 2–power order, then gi = ±I for each i, the number of i for which gi = Id is even (since the (−1)–eigenspace of g has square discriminant), and hence g ∈ E . In other words, E ∈ Syl2(A). A Sylow 2–subgroup of CSpin(E) is thus generated by E together with an element of order 2 which acts on E by inversion; this is an elementary abelian subgroup of rank 4, and is necessarily of type II.

We also need some more precise information about the subgroups of Spin7(q) of q rank 4 and type II. Let ψ ∈ Aut(Spin7(Fq)) denote the automorphism induced q by the ﬁeld automorphism (x 7→ x ). By Lemma A.3, Spin7(q) is precisely the subgroup of elements ﬁxed by ψq .

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Proposition A.9 Fix an odd prime power q, and let z ∈ Z(Spin7(q)) be the central involution. Let C and C′ denote the two conjugacy classes of subgroups E ≤ Spin7(q) of rank 4 and type I. Then the following hold.

−1 (a) For each E ∈E4 , there is an element a ∈ Spin7(Fq) such that aEa ∈C . For any such a, if we set

def −1 q xC(E) = a ψ (a),

then xC(E) ∈ E and is independent of the choice of a. ′ (b) E ∈C if and only if xC(E) = 1, and E ∈C if and only if xC (E)= z. II (c) Assume E ∈E4 , and set τ(E)= hz,xC (E)i. Then rk(τ(E)) = 2, and Aut (E)= α ∈ Aut(E) α| = Id . Spin7(q) τ(E) The four eigenspaces of E contained in the (−1)–eigenspace of xC(E) all have nonsquare discriminant, and the other three eigenspaces all have square discriminant.

2 Proof (a) For all E ∈ E4 , E has type I as a subgroup of Spin7(q ) since all elements of Fq are squares in Fq2 . Hence by Proposition A.8(b), for all ′ 2 4 −1 ′ E ∈ C , there is a ∈ SO7(q ) ≤ Ω7(q ) such that aEa = E . Upon lifting a 4 −1 to a ∈ Spin7(q ), this proves that there is a ∈ Spin7(Fq) such that aEa ∈C . Fix any such a, and set −1 q x = xC(E)= a ψ (a). q q −1 −1 −1 For all g ∈ E , ψ (g) = g and ψ (aga )= aga since E, aEa ≤ Spin7(q), and hence aga−1 = ψq(a)·g·ψq(a−1)= a(xgx−1)a−1.

Thus, x ∈ C F (E), and so x ∈ E since it is self centralizing in each Spin7( q) k Spin7(q ) (Proposition A.8(a)).

We next check that xC(E) is independent of the choice of a. Assume a, b ∈ −1 −1 Spin7(Fq) are such that aEa ∈ C and bEb ∈ C . Then by Proposition −1 −1 ′ −1 A.8(b), there is g ∈ Spin7(q) such that gbE(gb) = aEa . Set E = aEa ∈ −1 ′ ′ C , then gba ∈ N F (E ). Furthermore, since AutSpin (q)(E ) contains Spin7( q) 7 all automorphisms which send z to itself, and since E′ is self centralizing in k each of the groups Spin7(q ) (both by Proposition A.8 again), we see that ′ −1 q −1 N F (E ) is contained in Spin7(q). Thus, ba ∈ Spin7(q), so ψ (ba )= Spin7( q)

Geometry & Topology, Volume 6 (2002) 980 Ran Levi and Bob Oliver

−1 −1 q −1 q ba ; and this proves that xC(E)= a ψ (a)= b ψ (b) is independent of the choice of a.

(b) If E ∈C , then we can choose a = 1, and so xC(E)=1. ′ 2 If E ∈ C , then by Proposition A.8(b), there is a ∈ Spin7(q ) such that a ∈ −1 q SO7(q)rΩ7(q) and aEa ∈ C . Then ψ (a) 6= a since a∈ / Spin7(q) (Proposi- q −1 q tion A.3), but ψ (a) = a since a ∈ SO7(q). Thus, xC(E) = a ψ (a) = z in this case.

We have now shown that xC(E) ∈ hzi if E has type I, and it remains to prove −1 the converse. Fix a ∈ Spin7(Fq) such that aEa ∈ C . If xC(E) ∈ hzi, then q q ψ (a) ∈ {a, za}, so ψ (a) = a, and hence a ∈ SO7(q). Conjugation by an element of SO7(q) sends eigenspaces with square discriminant to eigenspaces with square discriminant, so all eigenspaces of E must have square discriminant since all eigenspaces of aEa−1 do. Hence E has type I.

II (c) Now write Spin = Spin7(q) for short. Assume E ∈E4 , and set x = xC(E) and τ(E)= hz,xi. Then x∈ / hzi by (b), and thus τ(E) has rank 2. By (a) (the uniqueness of x having the given properties), each element of AutSpin(E) restricts to the identity on τ(E). We have already seen (Proposition A.8(c)) that there is an element x(E) ∈ E such that the image in Aut(E) of AutSpin(E) is the group of automorphisms which fix x(E), and this shows that x(E)= x: the image in E of x. Since we already showed (Proposition A.8(c) again) that AutSpin(E) contains all automorphisms which are the identity on τ(E) and the identity modulo hzi, this finishes the proof that AutSpin(E) is the group of all automorphisms which are the identity on τ(E). The last statement (about the discriminants of the eigenspaces) follows directly from the first statement of Proposition A.8(c).

Throughout the rest of the section, we collect some more technical results which will be needed in Sections 2 and 4.

k−1 k Lemma A.10 Fix k ≥ 2. Let A = e13(2 ) ∈ GL3(Z/2 ) be the elementary k−1 matrix which has oﬀ diagonal entry 2 in position (1, 3). Let T1 and T2 be the two maximal parabolic subgroups of GL3(2): 1 T1 = GL2(Z/2) = (aij) ∈ GL3(2) | a21 = a31 = 0 and 2 T2 = GL1(Z/2) = (aij) ∈ GL3(2) | a31 = a32 = 0 . Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 981

Set T0 = T1 ∩ T2 : the group of upper triangular matrices in GL3(2). Assume that k µi : Ti −−−−−→ SL3(Z/2 ) are lifts of the inclusions (for i = 1, 2) such that µ1|T0 = µ2|T0 . Then there is a homomorphism k µ: GL3(2) −−−→ SL3(Z/2 ) such that µ|T1 = µ1 , and either µ|T2 = µ2 , or µ|T2 = cA ◦ µ2 .

′ k Proof We first claim that any two liftings σ, σ : T2 −−→ SL3(Z/2 ) are con- k jugate by an element of SL3(Z/2 ). This clearly holds when k =1, and so we ′ k−1 0 can assume inductively that σ ≡ σ (mod 2 ). Let M3 (F2) be the group of 0 3 × 3 matrices of trace zero, and define ρ: T2 −−→ M3 (F2) via the formula σ′(B) = (I + 2k−1ρ(B))·σ(B) 1 0 for B ∈ T2 . Then ρ is a 1–cocycle. Also, H (T2; M3 (F2)) = 0 by [9, Lemma 4.3] 0 (the module is F2[T2]–projective), so ρ is the coboundary of some X ∈ M3 (F2), and σ and σ′ differ by conjugation by I + 2k−1X .

By [9, Theorem 4.1], there exists a section µ deﬁned on GL3(2) such that k µ|T1 = µ1 . Let B ∈ SL3(Z/2 ) be such that µ|T2 = cB ◦ µ2 . Since µ|T0 = µ2|T0 , B must commute with all elements in µ(T0), and one easily checks that the k−1 only such elements are A = e13(2 ) and the identity.

Recall that a p–subgroup P of a ﬁnite group G is p–radical if NG(P )/P is p– reduced; ie, if Op(NG(P )/P ) = 1. (Here, Op(−) denotes the largest normal p– subgroup.) We say here that P is Fp(G)–radical if OutG(P ) (= OutFp(G)(P )) is p–reduced. In Section 4, some information will be needed involving the F2(Spin7(q))–radical subgroups of Spin7(q) which are also 2–centric. We ﬁrst note the following general result.

Lemma A.11 Fix a ﬁnite group G and a prime p. Then the following hold for any p–subgroup P ≤ G which is p–centric and Fp(G)–radical.

(a) If G = G1 × G2 , then P = P1 × P2 , where Pi is p–centric in Gi and Fp(Gi)–radical.

(b) If P ≤ H ⊳ G, then P is p–centric in H and Fp(H)–radical.

Geometry & Topology, Volume 6 (2002) 982 Ran Levi and Bob Oliver

(d) If G ⊳ G has p–power index, then P = G ∩ P for some P ≤ G which is p–centric in G and Fp(G)–radical. (e) If Q ⊳ G is a central p–subgroup, then Q ≤ P , and P/Q is p–centric in G/Q and Fp(G/Q)–radical. α (f) If G −−։ G is an epimorphism such that Ker(α) ≤ Z(G), then α−1(P ) is p–centric in G and Fp(G)–radical. e e e e Proof Point (a) follows from [16, Proposition 1.6(ii)]: P = P1 ×P2 for Pi ≤ Gi since P is p–radical, and Pi must be p–centric in Gi and Fp(Gi)–radical since ∼ CG(P )= CG1 (P1) × CG2 (P2) and OutG(P ) = OutP1 (G1) × OutP2 (G2).

Point (b) holds since CH (P ) ≤ CG(P ) and Op(OutH (P )) ≤ Op(OutG(P )). It remains to prove the other four points.

(e) Fix a central p–subgroup Q ≤ Z(G). Then P ≥ Q, since otherwise 1 6= NQP (P )/P ≤ Op(NG(P )/P ). Also, P/Q is p–centric in G/Q, since otherwise there would be x ∈ GrP of p–power order such that

1 6= [cx] ∈ Ker OutG(P ) −−−→ OutG/Q(P/Q) × OutG(Q) ≤ Op(OutG(P )).

It remains only to prove that P/Q is Fp(G/Q)–radical, and to do this it suﬃces to show that ∼ OutG/Q(P/Q) = OutG(P ). Equivalently, since P/Q and P are p–centric, we must show that

NG/Q(P/Q) ∼ NG(P ) ′ = ′ ; CG/Q(P/Q) × P/Q CG(P ) × P and this is clear once we have shown that ′ ∼ ′ CG/Q(P/Q) = CG(P ).

′ Any x ∈ CG/Q(P/Q) lifts to an element x ∈ G of order prime to p, whose conjugation action on P induces the identity on Q and on P/Q. By [15, Corollary 5.3.3], all such automorphisms of P have p–power order, and thus ′ x centralizes P . Since Q is a p–group and CG/Q(P/Q) has order prime to p, ′ this shows that the projection modulo Q sends CG/Q(P/Q) isomorphically to ′ CG(P ).

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α (f) Let G −−։ G be an epimorphism whose kernel is central. Clearly, α−1P −1 is p–centric in G. It remains only to prove that α P is Fp(G)–radical, and to do thise it suﬃces to show that e −1 ∼ e OutG(α P ) = OutG(P ). −1 Equivalently, since P and α (eP ) are p–centric, we must show that N (α−1P ) G ∼ NG(P ) ′ −1 −1 = ′ ; C (α P ) × α P CG(P ) × P G e and this is clear once we have shown that e C′ (α−1P ) ∼ C′ (P ). G = G This follows by exactly the same argument as in the proof of (e). e (c) Set P ′ = P ∩ H for short. Let

′ π ′ ′ ′ ′ NH (P ) −−−−−։ OutH (P ) =∼ NH (P )/(CH (P )·P ) be the natural projection, and set −1 ′ ′ K = π (Op(OutH (P ))) ≤ NH (P ). ′ ′ ′ ′ Then K ≥ Op(NH (P )) is an extension of CH (P )·P by Op(OutH (P )). It ′ ′ suffices to show that p ∤ [K:P ], since this implies that Op(OutH (P )) = 1 (ie, ′ ′ P is Fp(H)–radical), and that any Sylow p–subgroup of CH (P ) is contained in P ′ (hence P ′ is p–centric in H ). ′ ′ Assume otherwise: that p [K:P ]. Note first that P ⊳ NG(P ), and that N (P ) ≤ N (K); ie, N (P ) normalizes P ′ and K . The first statement is G G G obvious, and the second is verified by observing directly that NG(P ) normalizes ′ ′ NH (P ) and CH (P ). Thus the action of NG(P ) on K induces an action of ′ ′ NG(P ), and in particular of P , on K/P . Let K0/P denote the fixed subgroup of this action of P . Since p [K:P ′] by assumption, and since P is a p–group, p |K /P ′|. A straightforward check also shows that K ⊳ N (P ), and therefore 0 0 G that P K ⊳ N (P ). Also, since P ′ ≤ K ≤ H , 0 G 0 ′ P K0/P =∼ K0/(P ∩ K0)= K0/P is a normal subgroup of NG(P )/P of order a multiple of p. Since P is p–centric in G by assumption, ′ OutG(P )= NG(P )/(CG(P )·P )= NG(P )/(CG(P ) × P ), and hence the image of P K0/P in OutG(P ) is a normal subgroup which also has order a multiple of p.

Geometry & Topology, Volume 6 (2002) 984 Ran Levi and Bob Oliver

′ ′ By deﬁnition of K as an extension of CH (P )·P by a p–group, if x ∈ K has ′ order prime to p, then x ∈ CH (P ). Hence if x ∈ K0 has order prime to p, then for every z ∈ P , [x, z] ∈ P ′ , so x acts trivially on P/P ′ . Since x also centralizes ′ P , it follows that x centralizes P . This shows that the image of P K0/P in OutG(P ) is a p–group, thus a nontrivial normal p–subgroup of OutG(P ), and this contradicts the original assumption that P is Fp(G)–radical.

(d) Let G ⊳ G be a normal subgroup of p–power index and let P ≤ G be a p–centric and Fp(G)–radical subgroup. Let π ։ ∼ NG(P ) −−−−− OutG(P ) = NG(P ) CG(P )·P be the natural surjection, and set −1 K = π Op(OutG(P )) ≤ NG(P ). Then K is an extension of CG(P )·P by Op(OutG(P )). Fix any P ∈ Sylp(K). We will show that P ∩G = P , and that P is p–centric in G and Fp(G)–radical.

For each x ∈ K ∩ G ≤ NG(P ),

π(x) ∈ Op(OutG(P )) ∩ OutG(P ) ≤ Op(OutG(P )) = 1. Hence ∼ ′ x ∈ Ker NG(P ) −−−−→ OutG(P ) = (CG(P )·P ) ∩ G = CG(P ) · P = CG(P ) × P, ′ where CG(P ) ≤ CG(P ) is of order prime to p. Since the opposite inclusion is ′ obvious, this shows that K ∩ G = CG(P ) × P , and hence (since P ∈ Sylp(K)) that P ∩ G = P . ⊳ ′ Next, note that (K ∩G) K and K/(K ∩G) ≤ G/G, and hence K/CG(P ) has p–power order. Since P ∈ Sylp(K), P is an extension of P by K/(K ∩G), and ′ NK(P ) is an extension of a subgroup of (K ∩G) = (CG(P )×P ) by K/(K ∩G). ′ ′ Also, an element x ∈ CG(P ) normalizes P if and only if [x, P ] ∈ P ∩CG(P )=1. Hence ′ NK(P )= CK(P )·P = CG(P ) × P , (1) ′ ′ where CG(P )= CG(P )∩CG(P ) has order prime to p and is normal in NK(P ). ′ Since CG(P ) ≤ CG(P ) ≤ K , (1) shows that CG(P ) ≤ CG(P ) × P , and hence that P is p–centric in G. ⊳ It remains to show that P is Fp(G)–radical. Note ﬁrst that K NG(P ) −1 by construction, so for any x ∈ NG(P ), xPx ∈ Sylp(K). Since K is an

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′ ′ ⊳ extension of CG(P ) × P by the p–group K/(K ∩ G), and since CG(P ) K , it ′ follows that K is a split extension of CG(P ) by P . Hence for any x ∈ NG(P ), −1 −1 ′ xPx = yPy for some y ∈ CG(P ). Consequently, the restriction map ∼ ∼ NG(P )/CG(P ) = AutG(P ) −−−−−−→ AutG(P ) = NG(P )/CG(P ) (2) ∼ is surjective. Also, if x ∈ CG(P ) ≤ K normalizes P , then x ∈ NK(P ) = ′ P × CG(P ) by (1), and so cx ∈ Inn(P ). Thus the kernel of the map in (2) is contained in Inn(P ). Consequently,

∼ ∼ OutG(P ) = AutG(P )/ Inn(P ) = AutG(P )/ AutP (P ) = OutG(P )/Op(OutG(P )), and it follows that P is Fp(G)–radical.

This is now applied to show the following:

Proposition A.12 Fix an odd prime power q, and let P ≤ Spin7(q) be any subgroup which is 2–centric and F2(Spin7(q))–radical. Then P is centric in Spin7(Fq); ie, C F (P )= Z(P ). Spin7( q )

Proof Let z be the central involution in Spin7(q). By Lemma A.11(e), z ∈ P , def and P = P/hzi is 2–centric in Ω7(q) and is F2(Ω7(q))–radical. So by Lemma A.11(d), there is a 2–subgroup P ≤ O7(q) such that P ∩ Ω7(q)= P , and such that P is 2–centric in O7(q) and is F2(O7(q))–radical. b b Let V = m V be a maximal decomposition of V as an orthogonal direct b i=1 i sum of P –representations, and set b = b| . We assume these are arranged so L i Vi that for some k, dim(Vi) > 1 when i ≤ k and dim(Vi)=1 when i > k. Let V+ be the sumb of those 1–dimensional components Vi with square discriminant, and let V− be the sum of those 1–dimensional components Vi with nonsquare discriminant. We will be referring to the two decompositions

m k (V, b)= (Vi, bi)= (Vi, bi) ⊕ (V+, b+) ⊕ (V−, b−), Mi=1 Mi=1 both of which are orthogonal direct sums. We also write

(∞) (∞) V = Fq ⊗Fq V and Vi = Fq ⊗Fq Vi, b(∞) b(∞) and let and i be the induced quadratic forms.

Geometry & Topology, Volume 6 (2002) 986 Ran Levi and Bob Oliver

Step 1 For each i, set b Di = {± IdVi }≤ O(Vi, i), a subgroup of order 2; and write m D = Di ≤ O(V, b), and D± = Di ≤ O(V ±, b±). Yi=1 ViY⊆V± Thus D and D± are elementary abelian 2–groups of rank m and dim(V±), respectively. We ﬁrst claim that P ≥ D, (1) and that b m P = Pi where ∀ i, Pi is 2–centric in O(Vi, bi) and F2(O(Vi, bi))–radical. Yi=1 b (2) Clearly, [D, P ] = 1 (and D is a 2–group), so D ≤ P since P is 2–centric. This proves (1). The Vi are thus distinct (pairwise nonisomorphic) as P – representations,b since they are pairwise nonisomorphicb as D–representations.b The decomposition as a sum of Vi ’s is thus unique (not only up to isomorphism),b since HomP (Vi,Vj)=0 for i 6= j .

Let C be theb group of elements of O(V, b) which send each Vi to itself, and let N be the group of elements which permute the Vi . By the uniqueness of the decompositionb of V , b m P ·CO(V,b)(P ) ≤ C = O(Vi, bi) and NO(V,b)(P ) ≤ N. Yi=1 b b b b b Since P is 2–centric in O(V, b) and F2(O(V, b))–radical, it is also 2–centric in N and F2(N)–radical (this holds for any subgroup which contains NO(V,b)(P )). So byb Lemma A.11(b) (and since C ⊳ N ), P is 2–centric in C and F2(C)– radical.b Pointb (2) now follows from Lemma A.11(a). b b b b b b Step 2 Whenever dim(Vi) > 1 (ie, 1 ≤ i ≤ k), then by Lemma A.6, dim(Vi) is b b even, and i has square discriminant. So by Lemma A.4(a), − IdVi ∈ Ω(Vi, i) for such i. Together with (1), this shows that

k P = P ∩ Ω7(q) ≥ Di × Ω(V+, b+) ∩ D+ × Ω(V−, b−) ∩ D− . (3) i=1 Y b Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 987

Also, by Lemma A.4(a) again,

Ω(V±, b±) ∩ D± = SO(V±,b±) ∩ D± (4) = − IdVi⊕Vj k+1 ≤ i < j ≤ m, Vi,Vj ⊆ V± .

Step 3 By (3) and (4), the Vi are distinct as P –representations (not only as P –representations), except possibly when dim(V±) = 2. We ﬁrst check that this exceptional case cannot occur. If dim(V+) = 2 and its two irreducible summandsb are isomorphic as P –representations, then the image of P under b ′ projection to O(V+, +) is just {± IdV+ }. Hence we can write V+ = W ⊕ W , where W ⊥W ′ are 1–dimensional, P –invariant, and have nonsquare discriminant. Also, dim(V−) is odd, since V+ and the Vi for i ≤ k are all even dimensional. So − IdV−⊕W lies in CΩ7(q)(P ) but not in P . But this is impossible, since P is 2–centric in Ω7(q). The argument when dim(V−)=2 is similar.

The Vi are thus distinct as P –representations. So for all i 6= j , HomP (Vi,Vj)= 0, and hence (∞) (∞) ∼ Hom (V ,V ) = Fq ⊗Fq HomF (Vi,Vj) = 0. Fq[P ] i j q[P ] (∞) b(∞) (∞) Thus any element of O(V , ) which centralizes P sends each Vi to itself. In other words, m (∞) b(∞) C F (P )/hzi≤ C F (P ) ≤ O(V , ). Spin7( q) Ω7( q ) i i Yi=1

If dim(V±) ≥ 2, then since P contains all involutions in O(V±, b±) which are P –invariant and have even dimensional (−1)–eigenspace (see (3)), Lemma

A.4(c) shows that each element of Spin7(Fq) which commutes with P must act b on V± via ± Id. Also, for 1 ≤ i ≤ k, since − IdVi ∈ P by (3), each element in the centralizer of P acts on Vi with determinant 1 (Lemma A.4(c) again). We thus conclude that k (∞) b(∞) C F (P )/hzi≤ SO(Vi , i ) × {± IdV+ }×{± IdV− }. (5) Spin7( q) Yi=1 Step 4 We next show that k

C F (P )/hzi≤ {± IdVi }×{± IdV+ }×{± IdV− }. (6) Spin7( q ) Yi=1

Geometry & Topology, Volume 6 (2002) 988 Ran Levi and Bob Oliver

Using (5), this means showing, for each 1 ≤ i ≤ k, that

pri C F (P )/hzi ≤ {± IdVi }; (7) Spin7( q ) (∞) b(∞) (∞) b(∞) where pri denotes the projection of O7(Fq)= O(V , ) to O(Vi , i ). By Lemma A.6, dim(Vi) = 2 or 4. We consider these two cases separately.

′ def b Case 4A If dim(Vi) = 4, then by (2) and Lemma A.11(c), Pi = Pi ∩Ω(Vi, i) is 2–centric in Ω(Vi, bi) and is F2(Ω(Vi, bi))–radical. Also, by Proposition A.5, b ∼ + ∼ Ω(Vi, i) = Ω4 (q) = SL2(q) ×C2 SL2(q). ′ ′ By Lemma A.11(a,f), under this identiﬁcation, we have Pi = Q ×C2 Q , where ′ Q and Q are 2–centric in SL2(q) and F2(SL2(q))–radical. The Sylow 2– subgroups of SL2(q) are quaternion groups of order ≥ 8, all subgroups of a quaternion 2–group are quaternion or cyclic, and cyclic 2–subgroups of SL2(q) ′ cannot be both 2–centric and F2(SL2(q))–radical. So Q and Q must be quaternion of order ≥ 8. By [23, 3.6.3], any cyclic 2–subgroup of SL2(Fq) of order ≥ 4 is conjugate to a subgroup of diagonal matrices, whose centralizer is the group of all diagonal matrices in SL2(Fq). Knowing this, one easily checks that all nonabelian quaternion 2–subgroups of SL2(Fq) are centric in ′ SL2(Fq). It follows that Pi is centric in (∞) b(∞) ∼ SO(Vi , i ) = SL2(Fq) ×C2 SL2(Fq), and hence that ′ ′ ∞ ∞ pri C F (P )/hzi ≤ C ( ) b( ) (Pi )= Z(Pi )= {± IdVi }. Spin7( q) SO(Vi , i ) Thus (7) holds in this case. b ∼ ± Case 4B If dim(Vi) = 2, then O(Vi, i) = O2 (q) is a dihedral group of b order 2(q ∓ 1) [24, Theorem 11.4]. Hence Pi ∈ Syl2(O(Vi, i)), since the Sylow subgroups are the only radical 2–subgroups of a dihedral group. Fix Vj for any k < j ≤ m, and choose α ∈ O(Vi, bi) of determinant (−1) whose (−1)– b eigenspace has the same discriminant as Vj . Since Pi ∈ Syl2(O(Vi, i)), we can assume (after conjugating if necessary) that α ∈ Pi . Then (− IdVj ) ⊕ α lies in (∞) b(∞) P = P ∩ Ω7(q). Hence for any g ∈ C F (P )/hzi, pri(g) ∈ O(V , ) Spin7( q) i i leaves both eigenspaces of α invariant, and has determinant 1 by (5). Thus b pri(g)= ± IdVi ; and so (7) holds in this case. b Step 5 Clearly, − IdV± lies in SO(V±, ±) if and only if dim(V±) is even (which is the case for exactly one of the two spaces V± ), and this holds if and

Geometry & Topology, Volume 6 (2002) Construction of 2–local ﬁnite groups 989

b only if − IdV± ∈ Ω(V±, ±). Also, since each Vi for 1 ≤ i ≤ k has square b discriminant (Lemma A.6 again), − IdVi ∈ Ω(Vi, i) for all such i. Thus (6) and (1) imply that

C F (P )/hzi≤ P ∩ Ω7(q)= P , Spin7( q) and hence that P is centric in Spin7(Fq). b

Proposition A.12 does not hold in general if Spin7(−) is replaced by an arbitrary algebraic group. For example, assume q is an odd prime power, and let P ≤ SL5(q) be the group of diagonal matrices of 2–power order. Then P is 2–centric in SL5(q) and F2(SL5(q))–radical, but is deﬁnitely not 2–centric in SL5(Fq).

References

[1] M Aschbacher, A characterization of Chevalley groups over fields of odd order, Annals of Math. 106 (1977) 353–398 [2] M Aschbacher, Finite group theory, Cambridge Univ. Press (1986) [3] D Benson, Cohomology of sporadic groups, finite loop spaces, and the Dick- son invariants, Geometry and cohomology in group theory, London Math. Soc. Lecture notes ser. 252, Cambridge Univ. Press (1998) 10–23 [4] P Bousfield, D Kan, Homotopy limits, completions and localizations, Lecture notes in math. 304, Springer–Verlag (1972) [5] C Broto, R Levi, B Oliver, Homotopy equivalences of p–completed classifying spaces of finite groups, Invent. math. (to appear) [6] C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, preprint [7] C Broto, J Møller, Homotopy finite Chevalley versions of p–compact groups, (in preparation) [8] J Dieudonné, La géométrie des groupes classiques, Springer–Verlag (1963) [9] W Dwyer, C Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993) 37–64 [10] W Dwyer, C Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Annals of Math. 139 (1994) 395–442 [11] W Dwyer, C Wilkerson, The center of a p–compact group, The Cechˇ cen- tennial, Contemp. Math. 181 (1995) 119–157 [12] E Friedlander, Etale homotopy of simplicial schemes, Princeton Univ. Press (1982) [13] E Friedlander, G Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984) 347–361

Geometry & Topology, Volume 6 (2002) 990 Ran Levi and Bob Oliver

[14] D Goldschmidt, Strongly closed 2–subgroups of finite groups, Annals of Math. 102 (1975) 475–489 [15] D Gorenstein, Finite groups, Harper & Row (1968) [16] S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of BG via G–actions, Annals of Math. 135 (1992) 184–270 [17] J Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p–groupe abélien élémentaire, Publ. I.H.E.S. 75 (1992) [18] D Notbohm, On the 2–compact group DI(4), J. Reine Angew. Math. (to appear) [19] L Puig, Unpublished notes [20] L Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967) 58–93 [21] L Smith, Polynomial invariants of finite groups, AK Peters (1995) [22] R Solomon, Finite groups with Sylow 2–subgroups of type .3, J. Algebra 28 (1974) 182–198 [23] M Suzuki, Group theory I, Springer–Verlag (1982) [24] D Taylor, The geometry of the classical groups, Heldermann Verlag (1992) [25] C Weibel, An introduction to homological algebra, Cambridge Univ. Press (1994) [26] C Wilkerson, A primer on the Dickson invariants, Proc. Northwestern homotopy theory conference 1982, Contemp. Math. 19 (1983) 421–434

Geometry & Topology, Volume 9 (2005) ISSN 1364-0380 (on line) 1465-3060 (printed) 3001

Geometry & Topology Volume 9 (2005) 3001–21 (temporary page numbers) Erratum 1 Published: 9 February 2005

Correction to: Construction of 2–local finite groups of a type studied by Solomon and Benson Ran Levi Bob Oliver Department of Mathematical Sciences, University of Aberdeen Meston Building 339, Aberdeen AB24 3UE, UK and LAGA, Institut Galilée, Av. J-B Clément 93430 Villetaneuse, France Email: [email protected] and [email protected]

Abstract A p–local finite group is an algebraic structure with a classifying space which has many of the properties of p–completed classifying spaces of finite groups. In our paper [2], we constructed a family of 2–local finite groups which are “exotic” in the following sense: they are based on certain fusion systems over the Sylow 2–subgroup of Spin7(q) (q an odd prime power) shown by Solomon not to occur as the 2–fusion in any actual finite group. As predicted by Benson, the classifying spaces of these 2–local finite groups are very closely related to the Dwyer–Wilkerson space BDI(4). An error in our paper [2] was pointed out to us by Andy Chermak, and we correct that error here.

Keywords Classifying space, p–completion, ﬁnite groups, fusion

AMS Classiﬁcation 55R35; 55R37, 20D06, 20D20

A saturated fusion system over a finite p–group S is a category whose objects are the subgroups of S , whose morphisms are all monomorphisms between the subgroups, and which satisfy certain axioms first formulated by Puig, and also described at the start of the first section in [2]. The main result of [2] is the construction of saturated fusion systems over certain 2–groups, motivated by a theorem of Solomon [22], which implies that these systems cannot be induced by fusion in any finite group. Recently, Andy Chermak has pointed out to us that the fusion systems actually constructed in [2] are not saturated (do not satisfy all of Puig’s axioms). In this note, we describe how to modify that

c Geometry & Topology Publications 3002 Ran Levi and Bob Oliver construction in a way so as to obtain saturated fusion systems of the desired type, and explain why all of the results in [2] (aside from [2, Lemma A.10]) are true under this new construction. The following is the main theorem in [2]:

Theorem 1.2 [2, Theorem 2.1] Let q be an odd prime power, and ﬁx S ∈ Syl2(Spin7(q)). Let z ∈ Z(Spin7(q)) be the central element of order 2. Then there is a saturated fusion system F = FSol(q) which satisﬁes the following conditions:

(a) CF (z)= FS(Spin7(q)) as fusion systems over S . (b) All involutions of S are F –conjugate.

c Furthermore, there is a unique centric linking system L = LSol(q) associated to F .

We have, in fact, found two errors in our proof of this theorem which we correct here. The more serious one is in [2, Lemma A.10], which is not true as stated: the last sentence in its proof is wrong. This has several implications on the rest of our construction, all of which are systematically treated here. There is also an error in the statement of [2, Lemma 2.8(b)] which is corrected below (Lemma 1.9). We first state and prove here a corrected version of [2, Lemma A.10], and then state a modified version of the main technical proposition, [2, Proposition 1.2], used to prove saturation. Afterwards, we describe the changes which are needed in [2, Section 2] to prove the main theorem. In table 1, we list the correspondence between results and proofs in [2, Section 2] and those here. This is intended as a guide to the reader who is not yet familiar with [2], and who wants to read it simultaneously with this correction. The only difference between [2, Lemma A.10] and the corrected version shown here is that in [2], we claimed that the “correction factor” Z must lie in a certain k subgroup of order 2 in SL3(Z/2 ), which is definitely not the case. Also, for convenience, we state this lemma here for matrices over the 2–adic integers Z2 , instead of for matrices over the finite rings Z/2k . b Lemma 1.3 (Modified [2, Lemma A.10]) Let T1 and T2 be the two maximal parabolic subgroups of GL3(2): 1 T1 = GL2(Z/2) = (aij) ∈ GL3(2) | a21 = a31 = 0 Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local finite groups 3003

Reference in [2] Reference here Remarks Proposition 1.2 Proposition 1.4 more general statement Theorem 2.1 Theorem 1.2 unchanged Definition 2.2, Lemma 2.3 — unchanged Definition 2.4 — omit definition of Γn Prp. 2.5, Def. 2.6, Lem. 2.7 — unchanged — Lem. 1.5, Prp. 1.6 added — Definition 1.7 new definition of Γn — Lemma 1.8 added Lemma 2.8 Lemma 1.9 (b) restated, new proofs of (b), (e) Proposition 2.9 Proposition 1.10 partly new proof Lemma 2.10 Lemma 1.11 partly new proof Proposition 2.11 Proposition 1.12 partly new proof

Table 1 and 2 T2 = GL1(Z/2) = (aij) ∈ GL3(2) | a31 = a32 = 0 .

Set T0 = T1 ∩ T2 : the group of upper triangular matrices in GL 3(2). Assume, for some k ≥ 2, that µi : Ti −−−−−→ SL3(Z2) are lifts of the inclusions T −−→ GL (2) = SL (2) (for i = 1, 2) such that i 3 b3 µ1|T0 = µ2|T0 . Then there is a homomorphism

µ: GL3(2) −−−→ SL3(Z2), and an element Z ∈ C Z (µ1(T0)), such that µ|T = µ1 , and µ|T = cZ ◦ µ2 . SL3( 2) b 1 2 b ′ Proof By [1, Lemma 4.4], there is a lifting µ : GL3(2) → SL3(Z2) of the identity on GL3(2); and any two liftings to SL3(Z2) of the inclusion of T1 or of T2 into GL3(2) diﬀer by conjugation by an element of SL3(Z2). In particular,b ′ there are elements Z1,Z2 ∈ SL3(Z2) such that bµi = cZi ◦ µ |Ti (for i = 1, 2). ′ −1 Set µ = cZ1 ◦ µ , and Z = Z1Z2 . Then µ1 = µ|Ti , and µ|Tb2 = cZ ◦ µ2 . Since

µ1|T0 = µ2|T0 , conjugation by Z isb the identity on µ1(T0) = µ2(T0), and thus Z ∈ C (µ (T )). SL3(Z2) 1 0

b ⊳ Whenever G is a ﬁnite group, S ∈ Sylp(G), S0 S , and Γ ≤ Aut(S0), then

hFS(G); FS0 (Γ)i

Geometry & Topology, Volume 9 (2005) 3004 Ran Levi and Bob Oliver denotes the smallest fusion system over S which contains all G–fusion, and which also contains all restrictions of automorphisms in Γ. In other words, if F denotes this fusion system, then for each P,Q ≤ S , HomF (P,Q) is the set of all composites

ϕ1 ϕ2 ϕk P = P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−→ Pk = Q, where for each i, Pi ≤ S , and either ϕi ∈ HomG(Pi−1, Pi), or (if Pi−1, Pi ≤ S0 )

ϕi = ψi|Pi−1 for some ψi ∈ Γ such that ψi(Pi−1)= Pi .

Whenever G is a ﬁnite group and S ∈ Sylp(G), an automorphism ϕ ∈ Aut(S) is said to preserve G–fusion if for each P,Q ≤ S and each α ∈ Iso(P,Q), −1 α ∈ IsoG(P,Q) if and only if ϕαϕ ∈ IsoG(ϕ(P ), ϕ(Q)). The proof of Theorem 1.2 is based on the following proposition.

Proposition 1.4 (Modiﬁed [2, Proposition 1.2]) Fix a ﬁnite group G, a prime p dividing |G|, and a Sylow p–subgroup S ∈ Sylp(G). Fix a normal subgroup Z ⊳ G of order p, an elementary abelian subgroup U ⊳ S of rank two containing Z such that CS(U) ∈ Sylp(CG(U)), and a group Γ ≤ Aut(CS(U)) of automorphisms which preserve all CG(U)–fusion, and such that γ(U)= U for all γ ∈ Γ. Set

S0 = CS(U) and F = hFS(G); FS0 (Γ)i, and assume the following hold.

(a) All subgroups of order p in S diﬀerent from Z are G–conjugate. (b) Γ permutes transitively the subgroups of order p in U .

(c) {ϕ ∈ Γ | ϕ(Z)= Z} = AutNG(U)(CS(U)). (d) For each E ≤ S which is elementary abelian of rank three, contains U , and is fully centralized in FS(G),

{α ∈ AutF (CS(E)) | α(Z)= Z} = AutG(CS(E)). (e) For all E, E′ ≤ S which are elementary abelian of rank three and contain U , if E and E′ are Γ–conjugate, then they are G–conjugate.

Then F is a saturated fusion system over S . Also, for any P ≤ S such that Z ≤ P ,

{ϕ ∈ HomF (P,S) | ϕ(Z)= Z} = HomG(P,S). (1)

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3005

This proposition is slightly more general than [2, Proposition 1.2], in that Γ is assumed only to be a group of automorphisms of CS(U) which preserves CG(U)–fusion, and not a group of automorphisms of CG(U) itself. This extra generality is necessary when proving that the fusion systems FSol(q), under our modiﬁed construction, are saturated. The changes needed to prove this more general version of [2, Proposition 1.2] are described in Section 2.

Proposition 1.4 is applied with G = Spin7(q), Z = Z(G), and S ∈ Syl2(G), U ≤ S , and Γ ≤ Aut(CG(U)) to be chosen shortly. The error in [2] arose in the choice of Γ, as will be explained in detail below. We first recall some of the definitions and notation used in [2]. Throughout, we fix an odd prime power q, let Fq be a field with q elements, and let Fq be ∞ ∞ its algebraic closure. We write SL2(q ) = SL2(Fq), Spin7(q ) = Spin7(Fq), qn ∞ etc., for short. For each n, ψ denotes the automorphism of Spin7(q ) or of ∞ qn SL2(q ) induced by the field isomorphism (x 7→ x ). We then fix elements z, z1 ∈ Spin7(q) of order 2, where hzi = Z(Spin7(q)), set U = hz, z1i, and construct an explicit homomorphism ∞ 3 ∞ ω : SL2(q ) −−−−−→ Spin7(q ) such that

∞ Im(ω)= CSpin7(q )(U) and Ker(ω)= h(−I, −I, −I)i. Write ∞ def ∞ 3 H(q ) = ω(SL (q ) )= C ∞ (U) and [[X , X , X ]] = ω(X , X , X ) 2 Spin7(q ) 1 2 3 1 2 3 for short. In particular,

z = [[I,I, −I]] and z1 = [[−I,I,I]], and thus U = [[±I, ±I, ±I]] (with all combinations of signs). By [2, Lemma 2.3 & Proposition 2.5], there is an element τ ∈ N (U) of order 2 such that Spin7(q) −1 τ·[[X1, X2, X3]]·τ = [[X2, X1, X3]] ∞ for all X1, X2, X3 ∈ SL2(q ), and such that

∞ ∞ NSpin7(q )(U)= H(q )·hτi.

We next ﬁx elements A, B ∈ SL2(q) of order 4, such that hA, Bi =∼ Q8 (a quaternion group of order 8). Most of the following notation is taken from [2, Deﬁnition 2.6]. We set A = [[A, A, A]] and B = [[B,B,B]];

b b Geometry & Topology, Volume 9 (2005) 3006 Ran Levi and Bob Oliver

k ∞ ∞ 2 ∼ ∞ C(q )= {X ∈ CSL2(q )(A) | X = I, some k} = Z/2 ; and Q(q∞)= hC(q∞),Bi. Here, Z/2∞ means a union of cyclic 2–groups Z/2n for all n; equivalently, the 1 group Z[ 2 ]/Z. We then deﬁne A(q∞)= ω(C(q∞)3) =∼ (Z/2∞)3, ∞ ∞ 3 ∞ S0(q )= ω(Q(q ) ) ≤ H(q ) ∞ ∞ ∞ ∞ S(q )= S0(q )·hτi≤ H(q )·hτi≤ Spin7(q ).

∞ ∞ In all cases, whenever a subgroup Θ(q ) ≤ Spin7(q ) has been deﬁned, we set n ∞ n Θ(q )=Θ(q ) ∩ Spin7(q ). n qn ∞ Since Spin7(q ) is the ﬁxed subgroup of ψ acting on Spin7(q ) (cf. [2, Lemma A.3]), H(qn) ≤ H(q∞) is the subgroup of all elements of the form n qn [[X1, X2, X3]], where either Xi ∈ SL2(q ) for each i, or ψ (Xi) = −Xi for each i. By [2, Lemma 2.7], for all n, n n S(q ) ∈ Syl2(Spin7(q )).

The following lemma is what is needed to tell us how to choose a subgroup n n n n Γn ≤ Aut(S0(q )) so that the fusion system hFS(q )(Spin7(q )); FS0(q )(Γn)i is saturated. Note that since each element of C(q∞) has 2–power order, it makes u ∞ ∞ sense to write X ∈ C(q ) for X ∈ C(q ) and u ∈ Z2 .

∞ ∞ Lemma 1.5 Assume α ∈ Aut(A(q )) centralizes Autb S(q∞)(A(q )). Then α has the form v v u α([[X1, X2, X3]]) = [[X1 , X2 , X3 ]] ∗ for some u, v ∈ (Z2) .

Proof Set b ∞ ∆0 = AutS(q∞)(A(q )) = hc[[B,I,I]], c[[I,B,I]], c[[I,I,B]], cτ i for short. The second equality follows since S(q∞) is by deﬁnition generated by A(q∞) and the four elements listed. Set def ∼ 3 A1 = hz, z1, Ai = C2 , the 2–torsion subgroup in A(q∞). The image of ∆ ≤ Aut(A(q∞)) in the b 0 group Aut(A1) =∼ GL3(2) (the image under restriction) is the group of all

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3007

automorphisms which leave hzi and U = hz, z1i invariant (ie, the group of upper triangular matrices with respect to the ordered basis {z, z1, A}). −1 By assumption, [α, ∆0] = 1, and in particular, α∆0α = ∆0 .b Since each element of ∆0 sends U to itself, this means that each element of ∆0 also sends α(U) to itself. Also, U is the only subgroup of rank 2 left invariant by all elements of ∆0 (since ∆0 contains all automorphisms which leave hzi and U invariant), and hence α(U)= U . It follows that α induces an automorphism α′ of A(q∞)/U = (C(q∞)/h− Ii)3 . Also, α′ commutes with the following automorphisms of (C(q∞)/h− Ii)3 :

±1 ±1 ±1 (Y1,Y2,Y3) 7→ (Y1 ,Y2 ,Y3 ) and (Y1,Y2,Y3) 7→ (Y2,Y1,Y3) ′ since these are induced by automorphisms in ∆0 . Hence α has the form ′ v v u ∗ α (Y1,Y2,Y3) = (Y1 ,Y2 ,Y3 ) for some u, v ∈ (Z2) . Thus for all X1, X2, X3 ∈ C(q∞), v v u α([[X1, X2, X3]]) = [[±X1 , ±bX2 , ±X3 ]]. Since all elements of C(q∞) are squares, these signs must all be positive, and v v u α has the form α([[X1, X2, X3]] = [[X1 , X2 , X3 ]].

∗ ∞ For each u ∈ (Z2) , let δu ∈ Aut(A(q )) be the automorphism u b δu([[X1, X2, X3]]) = [X1, X2, X3 ]. ∞ Deﬁne γ, γu ∈ Aut(A(q )) by setting −1 γ([[X1, X2, X3]]) = [[X3, X1, X2]] and γu = δuγδu .

Proposition 1.6 There is an element u ∈ Z2 such that u ≡ 1 (mod 4), and ∞ such that the subgroup Ωu ≤ Aut(A(q )) given by b def ∞ Ω = h Aut ∞ (A(q )), γ i u Spin7(q ) u is isomorphic to C2 × GL3(2). Furthermore, the following hold:

(a) The subgroup of elements of Ωu which act via the identity on all 2–torsion in A(q∞) has order 2, and contains only the identity and the automorphism (g 7→ g−1). (b) For each n ≥ 1,

n n ∼ h AutSpin7(q )(A(q )), γui = C2 × GL3(2).

Geometry & Topology, Volume 9 (2005) 3008 Ran Levi and Bob Oliver

∞ k Proof For each k ≥ 1, let Ak ≤ A(q ) denote the 2 –torsion subgroup. In particular, ∼ 3 A1 = hz, z1, Ai = C2 , where A = [[A, A, A]]. ∞ ∞ Let R1,R2, ··· ∈ C(q ) =∼ Z/2 be elements such that R1 = −I , R2 = A, 2 b i b and (Ri) = Ri−1 for all k ≥ 2. Thus, |Ri| = 2 for all i. For each k ≥ 1, let (k) (k) (k) {r1 , r2 , r3 } be the basis of Ak deﬁned by (k) (k) (k) r1 = [[I,I,Rk]], r2 = [[Rk,I,I]], and r3 = [[Rk+1,Rk+1,Rk+1]]. (1) (1) (1) In particular, r1 = z, r2 = z1 , and r3 = A. Using these bases, we identify k ∞ Aut(Ak)= GL3(Z/2 ) and Aut(A(q )) = GL3(Z2). b Set b ∞ ∞ ∆ = Aut ∞ (A(q )), ∆ = Aut ∞ (A(q )), and ∆ = h∆ , γi. 0 S(q ) 1 Spin7(q ) 2 0

In particular, ∆2 is the group of all signed permutations ±1 ±1 ±1 [[X1, X2, X3]] 7→ [[Xσ(1), Xσ(2), Xσ(3)]] for σ ∈ Σ3 . (k) For each i = 0, 1, 2 and each k ≥ 1, let ∆i ≤ Aut(Ak) be the image of ∆i (1) under restriction. By [2, Proposition A.8], ∆1 = AutSpin(A1) is the group of ∼ (1) all elements of Aut(A1) = GL3(Z/2) which send z to itself. Also, ∆0 was seen in the proof of Lemma 1.5 to be the group of all automorphisms of A1 which leaves both z and U = hz, z1i invariant; and a similar argument shows that (1) ∆2 is the group of all automorphisms of A1 which leaves U invariant. Hence, (1) with respect to the ordered basis {z, z1, A} of A1 , each group ∆i ≤ Aut(A1) (i = 0, 1, 2) can be identiﬁed with the subgroup Ti ≤ GL3(Z/2) of Lemma 1.3. b By [2, Proposition 2.5],

∞ ∞ ∼ ∞ 3 CSpin7(q )(U)= H(q ) = (SL2(q ) )/h(−I, −I, −I)i.

∞ ∞ def An element [[X1, X2, X3]] ∈ H(q ) (Xi ∈ SL2(q )) centralizes A = [[A, A, A]] −1 −1 if and only if [Xi, A] = 1 for each i, or XiAXi = −A = A for each i. ∞ b Set C = CSL2(q )(A). This is an abelian group (the union of the ﬁnite cyclic

n ∞ groups CSL2(q )(A)), and NSL2(q )(A) = C·hBi. Hence, since A1 = hU, Ai and B = [[B,B,B]], we have b 3 C ∞ (A )= C ∞ (A)= ω(C )·hBi. (1) b Spin7(q ) 1 H(q )

b b Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3009

Since ω(C3) is abelian (thus centralizes A(q∞)), this shows that the kernel of ։ (1) each of the projection maps ∆i −− ∆i is generated by conjugation by B ; ie, by the automorphism (g 7→ g−1). b Since each ∆i is finite, their elements all have determinant of finite order in ∗ (1) (Z2) , hence are ±1 in all cases. Also, for each i, ∆i surjects onto ∆i = Ti with kernel generated by the automorphism (g 7→ g−1) of determinant (−1). Henceb the elements of determinant one in ∆i are sent isomorphically to Ti , and define a lift µi : Ti −−−−−−→ SL3(Z2) with respect to the given bases. In particular, µ | = µ | = µ . b1 T0 2 T0 0 ∗ ∞ For all i, j, k ∈ (Z2) , we define ψi,j,k ∈ Aut(A(q )) by setting i j k b ψi,j,k([[X1, X2, X3]]) = [[X1, X2 , X3 ]] ∞ for all [[X1, X2, X3]] ∈ A(q ). Recall that cτ ([[X1, X2, X3]]) = [[X2, X1, X3]] (by ∞ choice of τ ). Since ∆0 = AutS(q∞)(A(q )) is generated by c[[B,I,I]], c[[I,B,I]], ∞ ∞ c[[I,I,B]], and cτ (corresponding to generators of S(q )/A(q )), µ0 has image

Im(µ0)= µ1(T0)= hψ−1,−1,1, ψ1,−1,−1, cτ i, where ψ−1,−1,1 = c[[B,B,I]] and ψ1,−1,−1 = c[[I,B,B]]. By Lemma 1.3, there is a homomorphism

µ: GL3(2) −−−−−−→ SL3(Z2), and an element Z ∈ Aut(A(q∞)) =∼ GL (Z ) which commutes with all elements 3 2 b of µ1(T0), such that µ|T1 = µ1 and µ|T2 = cZ ◦ µ2 . By Lemma 1.5, Z = ψv,v,u ∗ (using the above notation) for some u, v ∈b (Z2) . Since ψv,v,v lies in the center of Aut(A(q∞)) (it sends every element to its v-th power), we can assume that v = 1 (without changing cZ ), and thus thatb Z = ψ1,1,u = δu . Finally, since δ−1 = ψ1,1,−1 ∈ ∆0 , we can replace δu by δ−u if necessary, and assume that u ≡ 1 (mod 4). ∞ Under the identiﬁcation GL3(Z2) = Aut(A(q )), we now have ∞ −1 Aut ∞ (A(q )) = (g 7→ g ) × µ(T1) Spin7(q )b and −1 0 1 0 γu = δuγδ = cZ (γ)= µ 1 1 0 , u 0 0 1 (1) (1) (1) where the matrix is that of γ|A1 with respect to the basis {r1 , r2 , r3 }. Also, T1 is a maximal subgroup of GL3(2) — the subgroup of invertible matrices

Geometry & Topology, Volume 9 (2005) 3010 Ran Levi and Bob Oliver

(1) which send r1 to itself — and so T1 together with this matrix generate GL3(2). Thus ∞ −1 Ω = h Aut ∞ (A(q )), γ i = hg 7→ g i×µ(GL (2)) ∼ C ×GL (2). (2) u Spin7(q ) u 3 = 2 3 This proves the ﬁrst claim in the proposition. Point (a) follows by construction, since each nonidentity element of µ(GL3(2)) acts nontrivially on A1 . Point (b) n follows from (2), once we know that each element of Spin7(q ) which normalizes n ∞ A1 (hence which normalizes A(q )) also normalizes A(q ) — and this follows from (1).

We are now in a position to deﬁne the fusion systems we want. Roughly, they n are generated by the fusion systems of Spin7(q ) together with one extra automorphism: the cyclic permutation [[X1, X2, X3]] 7→ [[X3, X1, X2]] “twisted” by the automorphism δu of the last proposition. By comparison, the construction in [2] was similar but without the twisting (ie, done with u = 1), and the resulting fusion system is, in fact, not saturated. We regard Q(q∞) = C(q∞)⋊hBi as an inﬁnite quaternion group: BA′B−1 = A′−1 for each A′ ∈ C(q∞), and each element of the coset C(q∞)·B has order 4. Hence any automorphism of C(q∞) extends to a unique automorphism of Q(q∞) which sends B to itself.

∗ Deﬁnition 1.7 Let u ∈ (Z2) be as in Proposition 1.6. Let γ, δ ∈ Aut(S (q∞)) b u 0 be the automorphisms b b γ([[X1, X2, X3]]) = [[X3, X1, X2]] and b ′ j ′ u j δu([[X1, X2, A B ]]) = [[X1, X2, (A ) B ]] ∞ ′ ∞ −1 for all Xi ∈ Q(q ), A ∈ C(q ), and i, j ∈ Z; and set γu = δuγδu ∈ ∞ b Aut(S0(q )). For each n ≥ 1, set n n b b bb Γn = h Inn(S0(q )), cτ , γui≤ Aut(S0(q )); and set n b n n n Fn = FSol(q )= hFS(q )(Spin7(q )), FS0(q )(Γn)i.

In order to be able to apply Proposition 1.4, it is important to know that Γn is fusion preserving. This follows immediately from the following lemma.

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3011

Lemma 1.8 For each n ≥ 1, the automorphisms cτ , γ , δu , and γu all preserve n n H(q )–fusion in S0(q ). b b b n Proof The fusion in SL2(q ) is generated by inner automorphisms of its Sylow subgroup Q(qn), together with the groups Aut(P ) for subgroups P ≤ Q(qn) isomorphic to Q8 . This follows from Alperin’s fusion theorem, since these are the only subgroups whose automorphism group is not a 2–group. Thus any n automorphism of Q(q ) — in particular, the automorphism ϕu deﬁned by ′ ′u ′ n ϕu(A )= A , ϕu(B)= B for A ∈ C(q ) — preserves fusion. n n 2n Set H0(q )= {[[X1, X2, X3]] | Xi ∈ SL2(q ), ∀i}. Fix a generator Y of C(q ); n 2 qn n then C(q ) = hY i, and ψ (Y ) = −Y . For each g = [[X1, X2, X3]] ∈ H(q ), qn qn qn [[ψ (X1), ψ (X2), ψ (X3)]] = [[X1, X2, X3]]; and hence there is some ﬁxed qn ǫ = ±1 for which ψ (Xi) = ǫ·Xi for i = 1, 2, 3. When ǫ = 1, this means n n that Xi ∈ SL2(q ) for each i; while if ǫ = −1 it means that Xi ∈ SL2(q )·Y n n for each i. So every element of H(q ) either lies in H0(q ), or has the form n g·[[Y,Y,Y ]] for some g ∈ H0(q ). n n −1 Assume g ∈ H(q ) and P,Q ≤ S0(q ) are such that gP g = Q. Clearly, n n n n P ≤ H0(q ) if and only if Q ≤ H0(q ) (H0(q ) is normal in H(q )). We claim that there is h ∈ H(qn) such that −1 hδu(P )h = δu(Q) and ch ◦ δu|P = δu ◦ cg|P . (1) 2n Let Pi,Qi ≤ SL2(q ) be the projections of P and Q to the i-th factor (i = b b b −1 b 1, 2, 3), and write g = [[X1, X2, X3]] (thus XiPiXi = Qi ). Consider the following cases.

n n (a) Assume g ∈ H0(q ) and P,Q ≤ H0(q ). Since ϕu preserves fusion in n n −1 SL2(q ), there is Y3 ∈ SL2(q ) such that Y3ϕu(P3)Y3 = ϕu(Q3) and def cY3 ◦ ϕu|P3 = ϕu ◦ cX3 |P3 . Then h = [[X1, X2,Y3]] satisfies (1). n n ′ (b) Assume g∈ / H0(q ) and P,Q ≤ H0(q ). Write g = g ·[[Y,Y,Y ]], where ′ n ′ g ∈ H0(q ). Choose h as in (a), so that (1) is satisfied with g, h replaced by g′, h′ . Then the element h = h′·[[Y,Y,Y u]] satisfies (1). n (c) Finally, assume that P,Q H0(q ). Then none of the subgroups Pi,Qi ≤ 2n n SL2(q ) is contained in SL2(q ). By the same procedure as was used in 2n (a), we can find h ∈ H0(q ) which satisfies (1); the problem is to do this so that h ∈ H(qn). 2n As noted above, fusion in SL2(q ) is generated by inner automorphisms 2n of Q(q ) and automorphisms of subgroups isomorphic to Q8 . Hence if Pi ′ 2n is not isomorphic to Q8 or one of its subgroups, then there is Xi ∈ Q(q )

Geometry & Topology, Volume 9 (2005) 3012 Ran Levi and Bob Oliver

such that c | = c ′ | . If, on the other hand, P (and hence Q ) is Xi Pi Xi Pi i i r s isomorphic to Q8 or C4 , then Pi ≤ hA, Y Bi and Qi ≤ hA, Y Bi for ′ def k 2n some odd r, s ∈ Z, and we can choose k ∈ Z such that Xi = Y ∈ Q(q ) has the same conjugation action as Xi . Thus in all cases, we can write ′ ′′ ′ 2n ′′ 2n Xi = XiXi for some Xi ∈ Q(q ) and some Xi ∈ CSL2(q )·hY i(Pi). The subgroups of Q(q2n) which are centralized by elements in the coset n SL2(q )·Y are precisely the cyclic subgroups. (The quaternion subgroups 2n of order ≥ 8 are all centric in SL2(q ).) Hence we can choose elements ′′ ′′ ′′ n ′′ n Yi as follows: Yi = 1 if Xi ∈ SL2(q ), and Yi ∈ SL2(q )·Y and ′′ n centralizes ϕu(Pi) if Xi ∈ SL2(q )·Y . We now deﬁne ′ ′ ′ ′′ ′′ ′′ h = [[X1, X2, ϕu(X3)]]·[[Y1 ,Y2 ,Y3 ]]; then h ∈ H(qn) and satisﬁes (1).

n n This shows that δu preserves H(q )–fusion as an automorphism of S0(q ). n Also, γ and cτ preserve fusion, since both extend to automorphisms of H(q ); −1 and hence γu = δubγδu also preserves fusion. b qn ∞ Let ψ = ψb ∈ bAut(Spinbb 7(q )) be induced by the ﬁeld automorphism x 7→ qn n x . By [2, Proposition A.9(a)], if E ≤ Spin7(q ) is an arbitrary elementary ∞ abelian 2–subgroup of rank 4, then there is an element a ∈ Spin7(q ) such that −1 aEa = E∗ , and we deﬁne −1 xC(E)= a ψ(a).

Then xC(E) ∈ E , and is independent of the choice of a. In the following lemma, we correct the statement and proof of points (b) and (e). The proof of (e) is aﬀected by both the changes in the statement of (b) and those in the deﬁnition of Γn .

n Lemma 1.9 [2, Lemma 2.8] Fix n ≥ 1, set E∗ = hz, z1, A, Bi ≤ S(q ), n U and let C be the Spin7(q )–conjugacy class of E∗ . Let E4 be the set of all n elementary abelian subgroups E ≤ S(q ) of rank 4 which containb bU = hz, z1i. n n Fix a generator X ∈ C(q ) (the 2–power torsion in CSL2(q )(A)), and choose Y ∈ C(q2n) such that Y 2 = X . Then the following hold.

(a) E∗ has type I. U (b) Each subgroup in E4 which contains A is of the form i j k Eijk = hz, z1, A, [[X B, X B, X B]]i bor ′ i j k E = hz, z1, A, [[X YB,X YB,X YB]]i. b ijk b Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3013

U n Each subgroup in E4 is H(q )–conjugate to one of these subgroups Eijk ′ or Eijk for some i, j, k ∈ Z. i j k ′ i j k (c) xC(Eijk) = [[(−I) , (−I) , (−I) ]] and xC(Eijk) = [[(−I) , (−I) , (−I) ]]·A. (d) All of the subgroups E′ have type II. The subgroup E has type I if ijk ijk b and only if i ≡ j (mod 2), and lies in C (is conjugate to E∗ ) if and only if i ≡ j ≡ k (mod 2). The subgroups E000 , E001 , and E100 thus represent the three conjugacy classes of rank four elementary abelian subgroups of n Spin7(q ) (and E∗ = E000 ). n U (e) For any ϕ ∈ Γn ≤ Aut(S0(q )) and any E ∈E4 , ϕ(xC (E)) = xC(ϕ(E)).

Proof We prove only points (b) and (e) here, and refer to [2] for the proofs of the other points.

(b) Assume first that A ∈ E ; ie, that E ≥ A1 = hz, z1, Ai. By definition, n n S(q ) is generated by A(q ), whose elements clearly centralize A1 ; and elements [[Bi,Bj,Bk]] for i, j, k ∈ {b0, 1}. Since an element of this formb centralizes A only if i = j = k, this shows that n b E ≤ CS(qn)(hz, z1, Ai)= A(q )·hBi. Since A(qn) is a finite abelian 2–group of rank 3, we have E = hz, z , A, gBi b b 1 for some g ∈ A(qn). Also,

n ∞ 3 n i j k i j k b b A(q )= ω(C(q ) ) ∩ Spin7(q )= [[X ,X ,X ]], [[X Y,X Y,X Y ]] i, j, k ∈ Z , ′ and hence E = Eijk or Eijk for some i, j, k ∈ Z. U Now let E ∈ E4 be arbitrary. Each element of E has the form [[X1, X2, X3]], n n where either Xi ∈ SL2(q ) for all i, or Xi ∈ SL2(q )·Y for all i — and the n elements of the ﬁrst type (Xi ∈ SL2(q )) form a subgroup of index at most 2. Since U has index 4 in E , this means that there is some g = [[X1, X2, X3]] ∈ n ErU for which Xi ∈ SL2(q ) for all i. Also, |Xi| = 4 for all i, since g∈ / n U = {[[±I, ±I, ±I]]}. Since all elements of order 4 in SL2(q ) are conjugate (cf. [4, 3.6.23]), this implies that g is H(qn)–conjugate to A, and hence that n ′ E is H(q )–conjugate to one of the above subgroups Eijk or Eijk . b (e) By construction, xC(−) is preserved under conjugation by elements of the n group Spin7(q ). Since Γn is generated by γu and conjugation by elements of n Spin7(q ), it suﬃces to prove the result when ϕ = γu . Since u ≡ 1 (mod 4) by ′ ′ ′ ′ ′ Proposition 1.6, γu(A) = A, γu(Eijk) = Ekb,i ,j , and γu(Eijk) = Ek′′,i′′,j′′ for ′ ′′ some i ≡ i ≡ i (mod 2), and similarly for the otherb indices. Hence by (c), ′ γu(xC(E)) = xC(γbu(Eb)) wheneverb b E = Eijk or E = Eijkb for some i, j, k ∈ Z. b b Geometry & Topology, Volume 9 (2005) 3014 Ran Levi and Bob Oliver

U ′ Now assume E ∈ E4 is not one of the subgroups Eijk or Eijk . By (b), there n ′ def −1 is g ∈ H(q ) such that E = gEg is of this form. Since γu preserves H(qn)–fusion by Lemma 1.8, there is h ∈ H(qn) such that the following square commutes: b γu E → γu(E) b cg c ↓ bh ↓ ′ γu ′ E → γu(E ) . ′ b ′ We have seen that γu(xC(E )) = xC(γu(E )); and also that cg and ch preserve xC(−). Hence γu(xC(E)) = xC(γu(E)) byb the commutativity of the square. b b The following isb the crucial resultb needed to apply Proposition 1.4. The statement is exactly the same as that in [2], but the proof has to be modiﬁed slightly due to the changed deﬁnition of Γn (hence of Fn ).

Proposition 1.10 [2, Proposition 2.9] Fix n ≥ 1. Let E ≤ S(qn) be an elementary abelian subgroup of rank 3 which contains U , and such that CS(qn)(E) ∈ Syl (C n (E)). Then 2 Spin7(q )

{ϕ ∈ Aut (C n (E)) | ϕ(z)= z} = Aut n (C n (E)). (1) Fn S(q ) Spin7(q ) S(q )

Proof Set n n Spin = Spin7(q ), S = S(q ), Γ=Γn, and F = Fn for short, and consider the subgroups

n def n n def R0 = R0(q ) = A(q ) and R1 = R1(q ) = CS(hU, Ai)= hR0, Bi. Then ∼ 3 b b R0 = (C2k ) and R1 = R0 ⋊ hBi, where 2k is the largest power which divides qn ± 1, and where B = [[B,B,B]] −1 b has order 2 and acts on R0 via (g 7→ g ). Also, ∼ 3 b hU, Ai = h[[±I, ±I, ±I]], [[A, A, A]]i = C2 is the 2–torsion subgroup of R0 . It was shown in the proof of [2, Proposition b 2.9] that 3 R0 is the only subgroup of S isomorphic to (C2k ) . (2)

Let E ≤ S be an elementary abelian subgroup of rank 3 which contains U , and such that CS(E) ∈ Syl2(CSpin(E)). There are two cases to consider: that where E ≤ R0 and that where E R0 .

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3015

Case 1 Assume E ≤ R0 . Since R0 is abelian of rank 3, we must have E = hU, Ai, the 2–torsion subgroup of R0 , and CS(E) = R1 . Also, by (2), neither R0 nor R1 is isomorphic to any other subgroup of S ; and hence b AutF (Ri)= AutSpin(Ri), AutΓ(Ri) = AutSpin(Ri), γu|Ri for i = 0, 1. (3)

n (Recall that AutΓ(Ri) is generated by AutS0(q )(Ri) and the restrictions of cτ and γu .) Hence by Proposition 1.6(b), ∼ AutF (R0)= AutSpin(R0), γu|R0 = C2 × GL3(2).

In other words, if we let A1 = hz, z1, Ai denote the 2–torsion subgroup of R0 , then restriction to A1 sends AutF (R0) onto Aut(A1) with kernel hcBi of order 2. Since AutSpin(A1) is the group of allb automorphisms of A1 which send z to itself [2, Proposition A.8], this shows that b

AutSpin(R0)= ϕ ∈ AutF (R0) ϕ(z)= z . (4) In the proof of [2, Proposition 2.9] (see formula (5) in that proof), we show the ﬁrst of the following two equalities:

AutSpin(R1) = Inn(R1)· ϕ ∈ AutSpin(R1) ϕ(B)= B

= Inn(R1)·ϕ ∈ Aut(R1) ϕ(B )= B, ϕ| R0 ∈ AutSpin(R0) . (5) b b The second equality holds since R1 = hR0b, Bi,b and since R0 is the unique abelian subgroup of R1 of index 2. Since γu(R0) = R0 and γu(B) = B, this, together with (3), shows that b b b b b AutF (R1) = Inn(R1)· ϕ ∈ Aut(R1) ϕ(B)= B, ϕ|R0 ∈ AutF (R0) , and hence that b b ϕ ∈ AutF (R1) ϕ(z)= z

= Inn(R1) · ϕ ∈ Aut( R1) ϕ(B)= B, ϕ(z)= z, ϕ|R0 ∈ AutF (R0)

= Inn(R1)·ϕ ∈ Aut(R1) ϕ(B)= B, ϕ|R0 ∈ AutSpin(R0) , (6) b b where the second equality follows from (4). If ϕ ∈ Aut(R ) is such that b b 1 ϕ(B) = B and ϕ|R0 = cx|R0 for some x ∈ NSpin(R0), then x normalizes R1 (the centralizer of the 2–torsion in R0 ), cx(B)= cy(B) for some y ∈ R1 = ∞ R0·hbBi byb (5), so we can assume y ∈ R0 . Since R0 = A(q ) is abelian − b b (so [y, R0] = 1), this implies cy 1x|R1 = ϕ, and hence ϕ ∈ AutSpin(R1). So {ϕ ∈bAutF (R1) | ϕ(z)= z}≤ AutSpin(R1) by (6), and the opposite inclusion is clear.

Geometry & Topology, Volume 9 (2005) 3016 Ran Levi and Bob Oliver

Case 2 Now assume that E R0 . By assumption, U ≤ E (hence E ≤ CS(E) ≤ CS(U)), and CS(E) is a Sylow subgroup of CSpin(E). Also, E i j k n contains an element of the form g·[[B ,B ,B ]] for g ∈ R0 = A(q ) and some n i, j, k not all even, and hence A(q ) CS(E). Hence by (2), CS(E) is not isomorphic to R1 = CS(hz, z1, Ai), and this shows that E is not Spin–conjugate to hz, z1, Ai. By [2, Proposition A.8], Spin contains exactly two conjugacy classes of rank 3 subgroups containingb z, and thus E must have type II. So by [2, Propositionb A.8(d)], CS(E) is elementary abelian of rank 4, and also has type II. n ∼ 4 Let C be the Spin7(q )–conjugacy class of the subgroup E∗ = hU, A, Bi = C2 , which by Lemma 1.9(a) has type I. Let E′ be the set of all subgroups of S which are elementary abelian of rank 4, contain U , and are not in C .b Byb Lemma ′ ′′ ′ ′ ′′ def ′ ′ 1.9(e), for any ϕ ∈ IsoΓ(E , E ) and any E ∈ E , E = ϕ(E ) ∈ E , and ϕ ′ ′′ ′ ′′ sends xC(E ) to xC(E ). The same holds for ϕ ∈ IsoSpin(E , E ) by deﬁnition ′ of the elements xC(−) ([2, Proposition A.9]). Since CS(E) ∈ E , this shows that all elements of AutF (CS(E)) send the element xC(CS (E)) to itself. By [2, Proposition A.9(c)], AutSpin(CS(E)) is the group of automorphisms which are the identity on the rank two subgroup hxC(CS(E)), zi; and (1) now follows.

The proof of the following lemma is essentially unchanged.

Lemma 1.11 [2, Lemma 2.10] Fix n ≥ 1, and let E, E′ ≤ S(qn) be two elementary abelian subgroups of rank three which contain U , and which are ′ n Γn –conjugate. Then E and E are Spin7(q )–conjugate.

Proof Consider the sets n 2 J1 = X ∈ SL2(q ) X = −I and 2n qn 2 J2 = X ∈ SL2(q ) ψ (X)= −X, X = −I . n n Here, as usual, ψq is induced by the ﬁeld automorphism (x 7→ xq ). It was shown in the proof of [2, Lemma 2.10] that all elements in each of these sets are SL2(q)–conjugate to each other. ′ Since E and E contain U , E, E ≤ C n (U). By [2, Proposition 2.5(a)], Spin7(q ) n def ∞ 3 n C n (U)= H(q ) = ω(SL (q ) ) ∩ Spin (q ). Spin7(q ) 2 7 Thus ′ ′ ′ ′ E = hz, z1, [[X1, X2, X3]]i and E = hz, z1, [[X1, X2, X3]]i,

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3017

′ ′ where the Xi are all in J1 or all in J2 , and similarly for the Xi . Also, E and E n 3 are Γn –conjugate, and each element of Γn leaves U = hz, z1i and ω(SL2(q ) ) ′ n 3 invariant. Hence either E and E are both contained in ω(SL2(q ) ), in which ′ n 3 case the Xi and Xi are all in J1 ; or neither is contained in ω(SL2(q ) ), in ′ ′ which case the Xi and Xi are all in J2 . This shows that the Xi and Xi are n ′ n all SL2(q )–conjugate, and so E and E are Spin7(q )–conjugate.

We are now ready to prove:

Proposition 1.12 [2, Proposition 2.11] For a fixed odd prime power q, let n ∞ ∞ ∞ S(q ) ≤ S(q ) ≤ Spin7(q ) be as defined above. Let z ∈ Z(Spin7(q )) be n the central element of order 2. Then for each n, Fn = FSol(q ) is saturated as a fusion system over S(qn), and satisfies the following conditions:

(a) For all P,Q ≤ S(qn) which contain z, if α ∈ Hom(P,Q) is such that

n α(z)= z, then α ∈ HomFn (P,Q) if and only if α ∈ HomSpin7(q )(P,Q). n n (b) CFn (z)= FS(qn)(Spin7(q )) as fusion systems over S(q ). n (c) All involutions of S(q ) are Fn –conjugate.

Furthermore, Fm ⊆ Fn for m|n. The union of the Fn is thus a category ∞ ∞ FSol(q ) whose objects are the ﬁnite subgroups of S(q ).

n n Proof We apply Proposition 1.4, where p = 2, G = Spin7(q ), S = S(q ), n n Z = hzi = Z(G), U = hz, z1i, CG(U) = H(q ), S0 = CS(U) = S0(q ), and n Γ=Γn ≤ Aut(S0). By Lemma 1.8, γu preserves H(q )–fusion in S0 . Since n Γ is generated by γu and certain automorphisms of H(q ), this shows that all n automorphisms in Γ preserve H(q )–fusion.b b Condition (a) in Proposition 1.4 (all noncentral involutions in G are conjugate) holds since all subgroups in E2 are conjugate ([2, Proposition A.8]), and condition (b) holds by deﬁnition of Γ. Condition (c) holds since

n n {γ ∈ Γ | γ(z)= z} = Inn(S0(q ))·hcτ i = AutNG(U)(S0(q )) by [2, Proposition 2.5(b)]. Condition (d) was shown in Proposition 1.10, and condition (e) in Lemma 1.11. So by Proposition 1.4, Fn is a saturated fusion n system, and CFn (Z)= FS(qn)(Spin7(q )). The proofs of the other statements remain unchanged.

Geometry & Topology, Volume 9 (2005) 3018 Ran Levi and Bob Oliver

In Section 3 of [2], these corrections affect only the proof of Lemma 3.1. In that proof, the groups E100 and E001 are not Γ1 –conjugate under the new definitions; instead E100 is Γ1 –conjugate to a subgroup which is Spin7(q)– conjugate to E001 (and hence the two are F –conjugate). Also, when showing that AutF (E001) is the group of all automorphisms which fix z = xC(E001), it is important to know that all Γ1 –isomorphisms between subgroups in that conjugacy class preserve the elements xC(−) (as shown in Lemma 1.9(e)), and not just that Γ1 –automorphisms of E001 do so. The changes do not affect the later sections. Just note that we are able to n kn consider FSol(q ) as a subcategory of FSol(q ) for k > 1, because they were ∗ both chosen using the same “correction factor” u ∈ (Z2) in the definitions of Γn and Γkn . b 2 Proof of Proposition 1.4

Proposition 1.4 follows from Lemmas 1.3, 1.4, and 1.5 in [2], once they are restated to assume the hypotheses of this new proposition, and not those of [2, Proposition 1.2]. The only one of these lemmas whose proof is aﬀected by the change in hypotheses is Lemma 1.4, and so we restate and reprove it here.

Lemma 2.1 Assume the hypotheses of Proposition 1.4, and let

F = hFS(G); FS0 (Γ)i be the fusion system generated by G and Γ. Then for all P, P ′ ≤ S which contain Z , ′ ′ {ϕ ∈ HomF (P, P ) | ϕ(Z)= Z} = HomG(P, P ).

Proof Upon replacing P ′ by ϕ(P ) ≤ P ′ , we can assume that ϕ is an isomorphism, and thus that it factors as a composite of isomorphisms

ϕ1 ϕ2 ϕ3 ϕk−1 ϕk ′ P = P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−→ Pk = P , =∼ =∼ =∼ =∼ =∼ where for each i, ϕi ∈ HomG(Pi−1, Pi) or ϕi ∈ HomΓ(Pi−1, Pi). Let Zi ≤ Z(Pi) be the subgroups of order p such that Z0 = Zk = Z and Zi = ϕi(Zi−1). To simplify the discussion, we say that a morphism in F is of type (G) if it is given by conjugation by an element of G, and of type (Γ) if it is the restriction of an automorphism in Γ. More generally, we say that a morphism is of type (G, Γ) if it is the composite of a morphism of type (G) followed by one of type

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3019

(Γ), etc. We regard IdP , for all P ≤ S , to be of both types, even if P S0 . By deﬁnition, if any nonidentity isomorphism is of type (Γ), then its source and image are both contained in S0 = CS(U). In particular, if Pi S0 for any 0

For each i, using [2, Lemma 1.3], choose some ψi ∈ HomF (PiU, S) such that ψi(Zi) = Z . More precisely, using points (1) and (2) in [2, Lemma 1.3], we can choose ψi to be of type (Γ) if Zi ≤ U (the inclusion if Zi = Z ), and to ′ be of type (G, Γ) if Zi U . Set Pi = ψi(Pi). To keep track of the eﬀect of morphisms on the subgroups Zi , we write them as morphisms between pairs, as shown below. Thus, ϕ factors as a composite of isomorphisms − ψ 1 ′ i−1 ϕi ψi ′ (Pi−1,Z) −−−−−→ (Pi−1,Zi−1) −−−−−→ (Pi,Zi) −−−−−→ (Pi ,Z).

ϕ1 ϕ2 ϕ3 (P0,Z) −−−−→ (P1,Z1) −−−−→ (P2,Z2) −−−−→ (P3,Z). (Γ) (G) (Γ)

Then Z1 = Z if and only if Z2 = Z because ϕ2 is of type (G). If Z1 = Z2 = Z , then ϕ1 and ϕ3 are of type (G) by Proposition 1.4(c), and the result follows.

If Z1 6= Z 6= Z2 , then U = ZZ1 = ZZ2 , and thus ϕ2(U) = U . Neither ϕ1 nor ϕ3 can be the identity, so Pi ≤ S0 = CS(U) for all i by deﬁnition of

HomΓ(−, −). Let γ1, γ3 ∈ Γ be such that ϕi = γi|Pi−1 , and let g ∈ NG(U) be −1 such that ϕ2 = cg . Then gCS(U)g ∈ Sylp(CG(U)), so there is h ∈ CG(U) such that hg ∈ N(CS(U)). Then chg ∈ Γ by Proposition 1.4(c). ′ Since γ3 preserves CG(U)–fusion among subgroups of S0 , there is g ∈ CG(U) ′ −1 −1 such that cg = γ3 ◦ ch ◦ γ3 . Thus ϕ is the composite c ′ ϕ1 chg −1 γ3 −1 g P0 −−−−→ P1 −−−−→ hP2h −−−−→ γ3(hP2h ) −−−−→ P3. (Γ) (Γ) (Γ) (G) The composite of the ﬁrst three isomorphisms is of type (Γ) and sends Z to g′−1Zg′ = Z , hence is of type (G) by Proposition 1.4(c) again, and so ϕ is also of type (G).

Geometry & Topology, Volume 9 (2005) 3020 Ran Levi and Bob Oliver

Case 2 Assume now that ϕ is of type (Γ, G, Γ, G, Γ); more precisely, that it is a composite of the form

ϕ1 ϕ2 ϕ3 (P0,Z) −−−→ (P1,Z1) −−−→ (P2,Z2) −−−→ (P3,Z3) (Γ) (G) (Γ)

ϕ4 ϕ5 −−−→ (P4,Z4) −−−→ (P5,Z), (G) (Γ)

where Z2,Z3 U . Then Z1,Z4 ≤ U and are distinct from Z , and the groups P0, P1, P4, P5 all contain U since ϕ1 and ϕ5 (being of type (Γ)) leave U invariant. In particular, P2 and P3 contain Z , since P1 and P4 do and ϕ2, ϕ4 are of type (G). We can also assume that U ≤ P2, P3 , since otherwise P2 ∩ U = Z or P3 ∩ U = Z , ϕ3(Z) = Z , and hence ϕ3 is of type (G) by Proposition 1.4(c) again. Finally, we assume that P2, P3 ≤ S0 = CS(U), since otherwise ϕ3 = Id.

Set H = CG(U) for short. Let E3 be the set of all elementary abelian subgroups ∗ E ≤ S of rank three which contain U , and let E3 ⊆E3 be the set of all E ∈E3 such that CS(E) ∈ Sylp(CG(E)). Then for all E ∈E3 ,

CS(E)= CS0 (E) and CG(E)= CH (E) ∗ since E ≥ U (and S0 = CS(U)). Thus E3 is the set of all subgroups E ∈ E3 which are fully centralized in the fusion system FS0 (H), and so each subgroup ∗ in E3 is H –conjugate to a subgroup in E3 .

Let γ1, γ3, γ5 ∈ Γ be such that ϕi = γi|Pi−1 for odd i. Let g2, g4 ∈ G be such that ϕi is conjugation by gi for i = 2, 4. We will construct a commutative diagram of the following form

γ1 cg2 γ3 cg4 γ5 (P0, E0) → (P1, E1) → (P2, E2) → (P3, E3) → (P4, E4) → (P5, E5)

c c c c c c a0 ↓ a1 ↓ a2 ↓ a3 ↓ a4 ↓ a5 ↓ c c ′ ′ γ1 ′ ′ h2 ′ ′ γ3 ′ ′ h4 ′ ′ γ5 ′ ′ (C0, E0) → (C1, E1) → (C2, E2) → (C3, E3) → (C4, E4) → (C5, E5) ′ ∗ ′ ′ where Ei ∈ E3 , Ci = CS(Ei), h2, h4 ∈ G, and ai ∈ H = CG(U). To do this, ′ ′ ′ ′ def ′ ′ −1 ∗ ﬁrst choose a0, a2, a4 ∈ H such that Ei = aiEiai ∈E3 for i = 0, 2, 4, and set

Geometry & Topology, Volume 9 (2005) Correction to: Construction of 2–local ﬁnite groups 3021

′ ′ ′ ′ Ei = γi(Ei−1) for i = 1, 3, 5. Then CS(Ei) = γi(CS(Ei−1)) for i = 1, 3, 5, so ′ ′ ′ ∗ ′ CS(Ei) ∈ Sylp(CH (Ei)) and Ei ∈E3 for all i. So we can choose x0 ∈ CH (E0) ′ ′ −1 −1 ′ ′ such that x0(a0P0a0 )x0 ≤ CS(E0), and set a0 = x0a0 ∈ H . Since γ1 ∈ Aut(S0) preserves H –fusion, there is a1 ∈ H which makes the ﬁrst square in ′ the above diagram commute. Now choose x2 ∈ CH (E2) such that ′ −1 ′ ′ −1 −1 −1 ′ x2 (a2g2a1 )CS(E1)(a2g2a1 ) x2 ≤ CS(E2), ′ −1 and set a2 = x2a2 and h2 = a2g2a1 . Upon continuing this procedure we obtain the above diagram. ′ ′ Let ϕ ∈ IsoF (CS(E0),CS (E5)) be the composite of the morphisms in the bot- tom row of the above diagram. Then ϕ(Z) = Z , since ϕ and the cai all ′ sendbZ to itself. By Proposition 1.4(e), the rank three subgroups Ei are all ′ −1 ′ G–conjugate to each other. Choose g ∈bG such that gE5g = E0 . Then ′ −1 ′ ′ g·CS(E5)·g and CS(E0) are both Sylow p–subgroups of CG(E0), so there is ′ ′ −1 ′ h ∈ CG(E0) such that (hg)CS (E5)(hg) = CS(E0). By Proposition 1.4(d), ′ chg ◦ ϕ ∈ AutF (CS(E0)) is of type (G), so ϕ is of type (G), and hence −1 ◦ ◦ ϕ = ca5 ϕ ca0 is also of type (G). b b b References

[1] W Dwyer, C Wilkerson, A new ﬁnite loop space at the prime two, J. Amer. Math. Soc. 6 (1993) 37–64 MR1161306 [2] R Levi, B Oliver, Construction of 2–local ﬁnite groups of a type studied by Solomon and Benson, Geom. Topol. 6 (2002) 917–990 MR1943386 [3] R Solomon, Finite groups with Sylow 2–subgroups of type 3, J. Algebra 28 (1974) 182–198 MR0344338 [4] M Suzuki, Group theory I, Grundlehren series 247, Springer–Verlag (1982) MR0648772

Geometry & Topology, Volume 9 (2005)